Hey, this is Brett and this 2019 Chevy Tahoe LT1 is stock number 13223Z. I am here at Summit Automotive in Fond du Lac, Wisconsin, your new and used SUV headquarters. This 2019 Chevrolet Tahoe has the 5.3 liter V8 gas engine. This vehicle has been fully safe and inspected by our service shop, has a fresh oil and filter change. Fluids have been checked and topped off and it is 100% ready to go. I'm going to go all the way around. In this video, inside, start it up, take a look at the back seats, take a look under the hood, and give you the most accurate representation that I can of the vehicle. Black clear coat is the color and I shoot all my videos in 4K. So if you have HD capabilities, turn them on now because it is the best way to check out the quality, condition, options, and of course, cleanliness of the vehicle before seeing it in person. And as you go down this side of the vehicle, you take note of how clean that body is. Didn't see any dents or dings on the rear quarter. Doors look really good. Paint is extremely reflective and shiny. Front fender is in great shape. If you like the video, you can subscribe to my YouTube channel, which is youtube.com slash summit auto click the bell notification so that updates on the videos I do each and every day of our new and used inventory. This one comes with the 20 inch polished aluminum rims and it has some really nice Michelin LTX AT tires. These are 275 55R20s and they have right around 50 to 60% of the tread left on them. Front fender once again, no dents or dings on there. It does come with the projector lamp headlamps, the LED running lights. Front bumper is in fantastic condition. Didn't see any dents, dings, scuffs or scrapes on there. And the grille is in really nice condition as well. Hood, absolutely perfect. Looks really good. Passengers side front fender is in excellent shape as well. And the passengers side front rim, no major scuffs or scrapes on there. Looks really good. As you go down this side of this 2019 Chevy Tahoe, once again take note of how clean that body is, how reflective and mirror like the paint is. I take these HD videos so if you are far away or even if you're close by and you just cannot make the trip down but you're still interested in purchasing the vehicle, you can see the vehicle, hear the vehicle and of confidence in what you're looking at before you even get here. So when you do get here, there's absolutely no surprises and you can make a smart and informed buying decision from wherever you're at. If this video helps you make that decision, let your salesman know that you saw the video that was helpful and that brat sent you back rooms in excellent shape as well. And the back tires have just as much tread as the front tires. Frame looks pretty good under here. This is a one owner, clean title history, clean car facts out of Maine. Back bumpers in excellent shape, no dents or dings on that, does have the back of parking sensors, does have the receiver hitch and seven pin wiring. Tailgate is in really nice shape. It is a power lift gate, so you just press the button, comes up like so. Back storage area is very nice and clean. These seats do fold down for extra storage and you can see just how nice and flat that they go. That second row folds down as well and then you have the strap to pull that back. You get storage underneath here and it does have a bag and then for your capless fuel fill, I think there might be a cargo net in there. So anyways, to get this gate down, you press this button right here, it comes on down. Does have a button for the flipper glass, which is nice to get into the back in case you have a trailer or a boat or something on, you can't open up the gate, you can still get to your stuff and you can see that those rear defrost cables are still attached. Coming around for full disclosure, this back rim is in nice shape as well, four wheel disc brakes of course on the Tahoe. We'll hop inside, check out the seats. Inside you get the black leather buckets in the front, both of these front seats are heated, no rips, no tears, they look really good. Both of them are power, factory floor mats, turned out four wheel drive auto headlamps, power pedals, stability control, lane departure warning, parking sensors, you can turn those off and on, tilt, telescopic steering wheel, power windows, locks and mirrors, wood grain trim, memory driver seat, Bose sound. Quick look at the back seats, they are just as clean as the front seats, no rips or tears, second row bench, so this is an eight passenger latch child safety system for any child car seats you may have. Back here you get rear air controls, it does have a 110 volt, 150 watt plug-in and a 12 volt power point, these seats fold down and up. To get to that third row, carpeting is very nice and clean. And the third row seats are in really nice condition as well, no rips or tears back there. It smells very clean inside this Tahoe, I don't think it's ever been smelted, and the inside of the doors and bottom of the doors are all very nice. We'll hop inside, check out the miles, the radio and everything that this Tahoe has to offer on the interior. You can see that this one has 67,559 miles, you do get a digital speedometer, compass display, instrument cluster is very nice and clean, comes with the leather wrapped steering wheel, Bluetooth and information center controls on the right, cruise controls and forward collision warning controls on the left, comes with the six speed automatic transmission with the optional tap shift, the Chevy MyLink system and FM and Sirius XM radio capabilities, and this one has the projection manager where you can project your cell phone to the screen via Android Auto or Apple CarPlay. So while this one doesn't have the factory navigation system, project your cell phone to that, and it's like you have Navra in the screen, and then you get a store away spot behind the radio with a USB jack, CD player, here are your climate controls, dual climate controls, heated seat buttons, has an aftermarket electric brake controller, passenger side format and seat are in excellent condition as well. And it does have the power sunroof. Up here you get sunroof controls, rear gate controls, home light, home light buttons for your garage doors, security systems and lane systems, map lights, on-star and SOS buttons in the mirror as well. We'll start it up and take a look under the hood. Starts right up, no check engine lights or anything like that, and we'll take a quick look under the hood. I would personally like to thank you for checking out the video today and hopefully from this HD video. You've been able to verify the quality condition options and of course the cleanliness of this Tahoe all the way around inside and out and under the hood. We have the 5.3 liter V8 gas engine. Engine bay is extremely clean, runs very smooth. Once again this Tahoe has been fully safety and inspected by our service shop as a fresh oil and filter change. All the fluids have been checked and topped off per the state of Wisconsin inspection process and it is 100% ready to go. There is the emission sticker and I would highly recommend this Tahoe for make quality and condition standpoint. I don't think you'll be upset with this one at all. Very happy with it. And to see more pictures of this vehicle or one of our other 550 new and used cars, trucks, SUVs, minivans, wranglers, half tons, three-quarter tons, one tons. You name it, we got to go to that website right there. SummitAuto.com, full pictures and descriptions of every single vehicle from two locations all at SummitAuto.com and if you'd like to check out more HD videos you can go to youtube.com slash SummitAuto. Remember to like, subscribe, and share on this video and all the videos that you see there. Click the bell notifications. You'll get updates on the videos I do each and every day here at Summit Automotive. In fact in a second you'll see a link to subscribe to my YouTube channel on the upper left. A link to more Tahoe videos I've done in the upper right. Link to this vehicle on our website on the lower left and a link to one of our latest YouTube videos in the lower right. Click those, check us out and we're super excited to help you with this Ultra Clean 2019 Chevrolet Tahoe LT1 in black clear coat. Thanks again for checking out the video. Remember to like, subscribe, and share on the YouTube channel. I really, really appreciate it. Thanks again.


So, today we are going to consider composite quadrature formulae and corrected trapezoidal rule. So, last time we had considered four basic rules, rectangle rule, then midpoint rule, trapezoidal rule and the Simpson's rule. For Simpson's rule, we saw that even though we are fitting a quadratic polynomial, there is no error if the function is a cubic polynomial. For the midpoint rule, we were fitting a constant polynomial, but we got exact attitude for polynomials of degree less than or equal to 1. Let me look at the comparison of the rules. So, for the rectangle, the degree of the interpolating polynomial was 0 and you got exact for polynomials of degree 0. This is expected for trapezoidal, it was 1 1 in both the cases. This is also expected, but for the midpoint degree of the interpolating polynomial was 0 and you had exact attitude for linear polynomials for Simpson quadratic polynomial was interpolated and we got exact attitudes for cubic polynomial. These trapezoidal rule and the Simpson's rule, they come under what are known as Newton codes formulae. So, you have interval a b, you subdivide it into n equal parts and then you consider these n plus 1 points. You fit a polynomial of degree less than or equal to n, you will get a formula and that is the Newton's codes formulae. So, if you choose n is equal to 1 and the two end points which gives us trapezoidal rule, if you choose n is equal to 3, two end points and the midpoint that gives us Simpson's rule. You can choose the n is equal to 3 and then you will get a formula of the form b minus a into 3 by 8 and then f x 0 plus 3 f x 1 plus 3 f x 2 plus f of x 3. You should note that if you add the coefficients 1 3 3 and 1, then these they will add up to 8. So, now you can choose general n and find a approximate quadrature formula, which we have seen Rangay's example, which was f x is equal to 1 upon 1 plus 25 x square for x belonging to minus 1 to 1. When we looked at the error between f and p n, then instead of the error being decreasing as n tends to infinity, the error tends to infinity. This is a classic example that the interpolating polynomials they may not converge in the maximum norm, that is why we went to piecewise polynomials. So, same thing we are going to do now that instead of integrating polynomials, we will integrate piecewise polynomials and that will give us the composite rules. Now, one of the disadvantage with piecewise polynomials is that they are not smooth functions. They lack differentiability properties at the partition point, but when you are integrating, then it does not matter. So, for the composite rules, we are going to get the good results. We will have convergence. The problem will come when we talk about numerical differentiation, but that little later. If you look at the Rangay's example, f x is equal to 1 upon 1 plus 25 x square x belonging to minus 1 to 1, then norm of f minus p n infinity norm tends to infinity and if you integrate p n x d x, then the error instead of decreasing, it will increase. So, now, we are going to consider the corrected trapezoidal rule. Suppose, you choose your interpolation points to be x 0 is equal to x 0 is equal to a and x 1 is equal to x 1 is equal to b.  So, this is our p 3 x.  So, both the polynomials, they are the same. When you integrate, you are going to get the same results. But, we have already done integrations for the trapezoidal rule, for the Simpson's rule. So, we want to make use of that. That is why I am taking my interpolation points as a b a b. So, the state remains the same. Whether you take them, the interpolation points as a a b b or whether you choose them as a b a b, they are going to be the same. So, now, the method is the same. In this case, the error is going to be f, the divided difference of f based on a a b b x minus a square plus a b b x multiplied by w x. w x is our product of x minus x j, where x j are interpolation points. In this case, our interpolation points are x 0 is equal to a repeated, x 1 x 0 is equal to a x 1 is equal to a x 2 is equal to b x 3 is equal to b. So, our derivative of w x becomes x minus a square x minus b square. So, w x is always going to be bigger than or equal to 0. The divided difference based on a repeated twice, b repeated twice and x is going to be continuous. So, we apply mean value theorem and then you work out the details and then one gets the formula. So, you have this first thing that is going to give us b minus a by 2 f a plus a b that was the trapezoidal rule. Then this f of a a b comes out of the integration sign. Integral x minus a into x minus b d x that we had already seen it to be equal to minus b minus a cube by 6. Then this will come out of the integration sign and you integrate this by parts. So, x minus b integration of x minus a by 2 minus a square. So, that will be x minus a cube by 3 evaluated between a and b minus integral a to b x minus a cube by 3 and the derivative of x minus b. So, that is x minus a cube by 3 d x. Then this is that b minus a by 2 f a plus a b term minus we make the adjustments and one can check that this comes out to be equal to minus divided difference of a based on a b minus f dash a divided by b minus a and then minus f dash b minus f of a b this 12 should be b minus a into b minus a cube by 12.  So, that is why it is called corrected trapezoidal rule. So, it is b minus a square by 12 f dash a minus f dash b and now for the error the divided difference is continuous this is bigger than or equal to 0 by mean value theorem for integrals this will come out as divided difference based on a a b b and some point c we want to integrate x minus a square x minus b square d x between a to b make the substitution as before y is equal to x minus a plus b by 2 and k to be equal to b minus a by 2 then x minus a is nothing but y plus k.  So, you will have y minus k square d y x varies between a to b. So, y will vary between minus b minus a by 2 to plus b minus a by 2.  So, we have fitted a polynomial of degree less than or equal to 3 we got the error to be exact for a polynomials of degree less than or equal to 3. So, if you look at the Simpsons method in Simpsons method we had 3 points f a f b and f of a plus b by 2 and the error was 0 for cubic polynomials. Here we had considered cubic hermit interpolation we got again the error to be 0 for cubic polynomials, but there is a catch of the error. We have the derivative values f dash a and f dash b coming into picture. So, if they are available well and good otherwise this formula is not applicable, but look at a case when f dash a is equal to f dash b.  So, for the trapezoidal rule we will have exactitude for polynomials of degree less than or equal to 3 and then afterwards we are going to consider what is known as Euler-Mclaurin series expansion and so on. So, if your function f is infinitely many times differentiable and the derivatives are periodic with period a b like this is the case for example, for sin x sin x has period 0 to 2 pi when you take their derivatives again they are periodic with period 0 to 2 pi. So, for such functions the trapezoidal rule is to be preferred, because then we are going to get very good convergence and the condition is your function should be periodic. So, function should be periodic and their derivatives also should have periodic property. So, this is for the basic rules. Now, we are going to consider composite rules. So, in the composite rules what we are going to do is divide interval a b into n equal parts. On each interval you apply some basic rule like look at the rectangle rule. So, on each interval you are going to apply rectangle rule and then we are interested in integral a to b f x dx. This integral will be equal to summation of integration over T i to T i plus 1 on the smaller interval and then we have got error formulae. So, you will add it up and then you will get an expression for error. The basic rectangle rule is integral a to b f x dx is approximately equal to f of a into b minus a and the error is f dash eta b minus a square by 2 where eta is some point in the interval a b. We have our interval T 0 to T n which is equal to a b. We have subdivided into smaller intervals and on the interval T i to T i plus 1 we apply the basic rule. So, we will have integral T i to T i plus 1 f x dx will be approximately equal to f of T i. The length of the interval b minus a is h. So, you will have h here and then h is T i plus 1 minus T i which is equal to b minus a by n. So, this is the error f dash eta h square by 2. Now, integral a to b f x dx is summation i goes from 0 to n minus 1 integral T i to T i plus 1 f x dx. This integral T i to T i plus 1 f x dx is approximately equal to value of f at the left hand point T i multiplied by length of the interval which is h and you add it up. So, you are going to have integral a to b f x dx to be summation i goes from 0 to n minus 1 f T i into h and the error will be f dash eta i h square by 2 summation i goes from 0 to n minus 1. So, it is h square by 2 and then this summation. So, look at this error it is h square by 2 summation i goes from 0 to n minus 1 f dash of T i h is b minus a by n. So, h square is going to tend to 0 as n tends to infinity, so h square is going to tend to 0 as n tends to infinity, but in this term as your n increases the number is going to increase. So, this is not something constant times h square by 2. Let us look at this error more carefully. Let me assume that f dash is continuous on interval a b.  So, m is going to be less than or equal to f dash T i less than or equal to m for i is equal to 0 1 up to n minus 1. So, if I look at summation i goes from 0 to n minus 1 f dash T i, this will be less than or equal to n times capital M and bigger than or equal to small n into small m, because we have got n terms. So, summation i goes from 0 to n minus 1 f dash T i divided by n will lie between small m and capital M. So, now, by the intermediate value theorem, this summation i goes from 0 to n minus 1 f dash T i divided by n will be equal to f dash c. For some c. So, here this error I keep h by 2 as it is h is b minus a by n. So, I will have b minus a and then summation i goes from 0 to n minus 1 f dash T i. So, this will be equal to h by 2 b minus a here it should be this divided by n and because there is h is b minus a by n and this is equal to f dash of c. So, we have got error to be equal to h by 2 b minus a f dash c. So, this is a constant independent of n and this will tend to 0 as n tends to infinity. So, our modulus of error will be less than or equal to constant times h, where constant is independent of n. So, we say that error is of the order of h. So, this is the error in the composite rectangle rule. Now, what we did for the rectangle rule we are going to do for our other rules that is the midpoint rule, then trapezoidal rule, then we had Simpson's rule and corrected trapezoidal rule. So, our interval a b is subdivided into small part. So, we have to do this. So, this is small intervals on each interval apply a basic rule. The integral a to b f x dx is split into n integrals, you get value you add it up. For the error also it will add it up. In the error there is going to be some power of h because in all the errors we had b minus a raise to something. Now, that b minus a is going to be replaced by h and then you will have summation, summation of n terms. So, that summation of n terms we get rid of by say adjoining 1 h to h. So, that is why you are going to have always 1 power of h to be less. In the sense, in the rectangle rule our error had b minus a square. So, corresponding error in the composite rectangle rule we had not h square, but only h. In the midpoint rule we had b minus a cube. So, that will be replaced by h square and then it is the exactly same argument. So, what I am going to do is write down the composite rule and then write down the error, but whatever we have worked out the details they are exactly the same. So, we are not going to work them out again. Look at the midpoint rule. Integral a to b fxdx is approximately equal to b minus a f of a plus b by 2. The error was f double dash eta b minus a cube by 24. When you consider the composite rule in the interval ti to ti plus 1 integral fxdx that is approximated by length of the interval which is h multiplied by value of f at ti plus ti plus 1 by 2. In the error we are going to have summation f double dash eta i h cube by 24. Then 1 h we take from 1 h n and then you have got this integral fxdx which is average. This average will be equal to f double dash of psi. So, you have got b minus a by 24 h square and then f double dash psi. You need the function to be 2 times differentiable and the error is going to be of the order of h square. Then when we look at the composite trapezoidal rule it is similar.  So, instead of b minus a cube by 24 you have got b minus a cube by 12 that is the only difference. Then you are going to have the error to be minus b minus a f double dash psi divided by 12 into h square.     So, this f ti is common. So, this is for trapezoidal rule. If you had considered midpoint rule you will be considering midpoint here, midpoint here. So, when you are looking at two intervals you need to evaluate the function at three points. Here you are you need to evaluate it at two points and in general when you have n intervals for midpoint rule you will have n points for trapezoidal rule you will have n plus 1 point.  So, at the two end points the weight is different that is h by 2 and at the interior partition points the weight is h and the error is constant times h square. The function evaluations are n plus 1 you need function to be two times differentiable and the error is of the order of h square. For the Simpson's rule we have got value of the function at two end points and value at the midpoint and the error was given by this formula which involved fourth derivative of the function and then b minus a by 2 raised to 5.  So, you have got weights to be different at the two end points a and b at the partition points t 1, t 2, t n minus 1 and at the midpoints of the intervals. Next we look at the error. So, the error will be given by this b minus a by 2 is replaced by h by 2 raised to 5 minus 1 by 90 and then this is from each interval you are going to have f 4 it i. So, that is why you have summation i goes from 0 to n minus 1. Now this summation we associate 1 h with it and then you get a 4 si into b minus a and then you have h by 2 raised to 4 and then divided by 180. So, this is the error in the Simpson's rule the function evaluations are 2 n plus 1 you need function to be 4 times differentiable and the error is of the order of h raised to 4 and now we look at the corrected trapping trapezoidal rule.    So, this is one term the other term will be b minus a square by 12 f dash a minus f dash b. So, you will have instead of b minus a you will have h square a by 2 and then you will have f dash t i minus f dash t i plus 1 that is from the interval t i to t i plus 1. If you consider the interval t i minus 1 to t i there it will be f dash t i minus 1 minus f dash t i. So, when you add it up f dash t i will get cancelled.    So, you have integral a to b f x d x to be equal to summation i goes from 0 to n minus 1 integral t i to t i plus 1 f x d x. This will be approximately equal to summation i goes from 0 to n minus 1 h by 2 f of t i plus f of t i plus 1 plus h square by 12 summation i goes from 0 to n minus 1 integral f x d x. So, this is our summation i goes from 0 to n minus 1 f dash t i minus f dash t i plus 1. So, this is our trapezoidal rule composite and this simplifies to h square by 12 f dash a minus f dash b. So, you are just you have to just add one extra term.  And if f dash a is equal to f dash b then this term will not be there and when you look at the error, error is going to be h raise to 5 divided by 720 and this summation it is summation of n terms. So, we associate one h with it and that will give us b minus a f 4 evaluated at some point psi and then you have h raise to 4 and 720. So, if the function f is 4 times continuously differentiable, the error is going to be of the order of h raise to 4, we need n plus 1 function evaluations that was from the composite trapezoidal rule and then we need to evaluate two derivative functions. So, here is the comparison for the rectangle rule the smoothness required is c 1 for the midpoint it was c 2 midpoint trapezoidal c 2 and Simpson and corrected trapezoidal c 4. The function evaluations in rectangle and midpoint they are the same n, but the order of convergence is h in rectangle and h square in midpoint. So, you should prefer midpoint over rectangle between midpoint and trapezoidal both need the function to be two times differentiable, the order of convergence is h square and the function evaluation is n here and n plus 1 in trapezoidal for n big enough there is not much difference between n and n plus 1. So, midpoint and trapezoidal they are on par for the Simpson you are getting the improvement in the order of convergence from h square to h raise to 4, but the function evaluations are instead of n plus 1 you have got two n plus 1. So, that means it is double, but this improvement is or this increase in the function evaluation it is arithmetic and the order of convergence that improvement is geometric. So, if your function is sufficiently differentiable Simpson's rule will be preferable compared to the trapezoidal rule and then corrected trapezoidal rule it will have the same function evaluation you will have two derivatives. So, if the derivative values at the two end points it is available then you should prefer corrected trapezoidal rule over all the other rules. So, we have looked at some basic rules and then we considered the corresponding composite rules and we have got way to find approximate value of the integration. We wanted to find the order of convergence that is why we had assumed certain differentiability properties of our function, but look at the rectangle rule in the rectangle rule when you consider the composite rectangle rule it is given by h times summation i goes from 0 to n minus 1 f at T i that means we have got our partition uniform partition of interval a b. In each interval we are choosing the left end point. So, we are looking at f of T i and then you are multiplying by h. So, that is nothing but the but a Riemann sum. So, if your function f is continuous then this composite rectangle rule h times summation f T i i goes from 0 to n minus 1 that is going to converge to integral a to b f x d x. So, this f to be continuously differentiable that we need to say that the order of convergence is h in the composite rule it is not necessary for the convergence. Look at the midpoint rule in the midpoint rule we are looking at the in each interval T i to T i plus 1 we are looking at the midpoint and then we are forming the Riemann sum. For the Riemann sum in the interval T i to T i plus 1 you can choose any point multiply by h add it up and then it is going to converge if your function is continuous. Now, what happens for the trapezoidal rule?   So, now the question is whether this argument about the Riemann sum whether we can apply it to the composite trapezoidal rule that suppose the function is given to be only continuous we do not know that it may not be differentiable in that case whether the composite trapezoidal rule is going to converge. So, let us try to write this composite trapezoidal rule as Riemann sums. So, we had the weight at the 2 n point was h by 2 and at the interior points it was h. So, let us look at the Riemann sum where you are choosing the left end point that means what points will come into picture will be a T 1 T 2 up to T n minus 1 and then you have to multiply by h. So, that will be one Riemann sum another Riemann sum will be choose the right end point in that case in the interval T 0 to T 1 we will be choosing f of T 1 that means a will not come into picture. So, you will have T 1 T 2 and T n is equal to b. So, here we have got h by 2 then h by 2 f a plus h by 2 summation i goes from 1 to n minus 1 f of T i. So, this is plus h by 2 summation i goes from 1 to n minus 1 f of T i plus h by 2 f of b. So, this will be equal to let me call R n 1 or R n plus s n divided by 2 where R n is summation f T i i goes from 0 to n minus 1 s n is summation f T i i goes from 1 to n. So, here in the Riemann sum you are choosing the left end point that gives us R n if you choose the right end point that will give us s n our R n will tend to integral a to b f x d x s n also will tend to integral a to b f x d x. So, this will tend to integral a to b f x d x. So, thus even for the composite trapezoidal rule we are going to have convergence under the condition that function is continuous. If you want to know the order of convergence then you assume f to be twice differentiable. Now, I will leave it to you to do it for Simpsons like look at the composite Simpsons rule and then try to write it as some Riemann sums and show that the Simpson rule is also convergent provided your function is continuous. So, just continuity is enough all these composite rules they are going to converge. The order of convergence that needs the function to be differentiable or more than once differentiable. Now, we are going to look at another important rule and that is known as Gaussian integration. So, the idea is so far what we have done is we have fixed our points x 0, x 1, x n look at the basic rules how we have calculated or how we have derived them. We have fixed x 0, x 1, x n some way then we fit a polynomial of degree less than or equal to n. You will know how to integrate. So, you integrate and then you get a rule. The rule which we got was of the type value of the function at interpolation points x i and then multiplied by some weight w i like look at the rectangle rule.  So, both f a and f b they had the same weight. So, now is it possible that we do not fix x 0, x 1, x n to be beforehand. We want a integration rule of the form summation w i f x i i goes from 0 to n. Treat the w i's and x i's as unknowns. That means try to determine w 0, w 1, w n x 0, x 1, x n in such a manner that we have error to be 0 for as high degree polynomial as possible. So, we have got w 0, w 1, w n x 0, x 1, x n. So, we have got 2 n plus 2 points to be determined. So, we have got x 1, x n to be determined. So, we can hope to have exactitude for polynomials of degree less than or equal to 2 n plus 1. So, this we are going to consider in our next lecture. Thank you.


In lecture one of this series, we talked about the idea of distance, where distance represents how much space, how much length is between two objects, right? And we use the distance formula to calculate that. That is a form of distance measurement, right? We can measure the distance, which might be called length, or width, or height. With those, the difference between those words is somewhat arbitrary. That kind of has to do with direction. Length and width usually represent some type of horizontal thing. Height often represents a vertical distance. Again, those are kind of just words that have sort of colloquial meanings, but they all represent distances. And how can we measure distance? Well, it turns out there's a lot of different ways of measuring distance. You can measure distance in feet. You can measure distance in inches. You can measure distance in miles, just to name a few. And it depends on what you're measuring to decide what you're going to do, right? If it's a very, very large distance, you might be measuring it in miles, like the distance between countries or cities or states. If you're measuring the distance on a more moderate scale, maybe you're measuring in feet, right? It's like, oh, these things are 200 feet away from each other. Or if things are relatively close, you might be using inches, or you might be using the idea of a yard, for example. And there's these different measurements of distance you one could use. Like for example, I had my son told me a joke earlier today that how do you measure a snake? How do you measure the length of a snake? And I was like, I don't know, how do you measure the length of the snake? And he's like, oh, you have to measure the length of a snake in inches because they don't have any feet. And so when it comes to measurements, it's really, it has a lot to do with relativness, like how close are they together. But there's also a lot of culture that's built into measurements. Because when it comes to feet, inches, yard, miles, why not use things like centimeters, or meters, or kilometers, or nanometers, or we can keep on going, right? There are some competing forces of competing measurement systems one could use. And so for those, for those people, generally speaking, who were raised in the United States, if you're measuring distance, you're quite comfortable with sort of this traditional British method. But many places in the world, with the exception of the United States, have migrated to more the SI units, SI units, which, because there's some benefits to them in scientific measurements, mostly the switch between the different measurements is much easier, because it's usually multiples of 10 or multiples of 1000 or things like that. And so for some people, if you're like, oh yeah, my house is five kilometers away. Some people have no idea how large five kilometers are.  I'm not saying those are equivalent to each other. But some people might have that frame of reference right there. And so measurements, the measurements we try to use have a lot to do with culture. And sometimes there are measurements that are introduced that have some more scientific advantage. But still a lot of it has to do with culture. And so some measurements might feel natural or standard or straightforward. But that's, you know, other ones might feel awkward or bizarre or unnatural. But those are really just our perceptions. There's nothing necessarily better about one or the other in terms of this is a better measurement foot, feet versus meters or something like that, right? Again, that has a lot to do with culture. The reason I talk about this is that angle measurement is the same basic idea. In this trigonometry series, there's two particular forms of angle measure that we're going to focus on. In this video, we're going to talk about the idea of degree measure. And sometime later on in this lecture series, we'll introduce a competing notion of angle measure called the radiant measure. And so let's first talk about the degree measure for a moment. At least in the United States, I would say that the degree measure is the most common, most comfortable, so to speak, most natural way to measure angles. But I want you to be aware that the choice of 360 degrees is somewhat of an arbitrary choice. But this is how we define angle measure. So we want to put a quantity to describe how big one angle is compared to another. And so in the degree measure, a degree is denoted by the drawing a little circle as a superscript here. Same thing like when we use temperature degrees, although it has a different meaning, same notation here. 360 degrees represents one complete rotation. So if you took, take a ray and rotate around in a whole circle, so that is the initial side and the terminal side are exactly the same thing. This represents a 360 degree measure. So there's 360 degrees in one complete rotation. So what this tells you is if there's 360 degrees in a full rotation, a half angle, which is half of a rotation, it's just a half plane, that would be half of 360 degrees. And that's going to be 180 degrees. A right angle, which is half of a flat angle, then, would then take half of that. A right angle would be considered 90 degrees. And that's a statement many of us are probably familiar with. A null angle, which would take no rotation whatsoever, you just start and stop at the same spot, that would then be zero degrees. And so some of these important angles we talked about earlier can be described using a number. How big is the angle? In all, it's based upon this observation that we're deciding that 360 degrees gives us one complete rotation. Some other vocabulary we should connect to that an angle is acute if it's between zero degrees and 90 degrees. An angle is obtuse if its angle measure is between 90 degrees and 180 degrees. And in particular, one could say that one degree is worth one 360th of a rotation. Why 360? What's so significant about that? Well, the 360 degree system, it actually dates back to the Babylonian culture, which existed roughly 4,000 years ago from the recording of this video. This is the same culture that introduced at least US American culture to the idea of 60 seconds in a minute. You have 60 minutes in an hour. 60 seconds equals one minute. 60 minutes equals one hour. And so they have 360, which is six times 60, of course, in a complete rotation. So hence, degrees are deeply embedded into many of our cultures. Even if it wasn't there historically, often it's just so used international, many of us adopted it because of that. Again, it's just so ancient and so heavily used that it feels natural because degree measure is essentially our native language when it comes to angle measurement. That doesn't mean it's necessarily the best or the easiest to use. It's just the one we have the most experience with. And so that's why it feels so familiar to most people. Another thing we should mention about degree measure is the difference between clockwise and counterclockwise rotation. When it comes to a degree measure, if we have our initial side and our terminal side, so our initial side right here, IS, our terminal side TS right here, if you take a counterclockwise rotation, this is considered a positive angle measurement. On the other hand, if you take your initial side here, and you rotate it here to your terminal side, and you take a clockwise rotation, that's considered a negative angle measure. So positive degrees, it's going to be counterclockwise. Negative degrees is going to be considered a clockwise rotation. Why is clockwise negative and counterclockwise positive? Well, again, clockwise versus counterclockwise, this is all just cultural things. There's no reason to think that clockwise is the preferred direction that a clock should spin. After all, clocks, which historically were circles, right? So as this is this our hand goes around, right? Why does it go that direction? Why not the other one? Well, the reason for this really just comes down to sundials. So you have a traditional sundial, right, which is a circle, which has a big triangle on it, which then cast a shadow on the sundial based upon the location of the sun. Well, because of the rotation of the earth, the shadow of the sundial would move around in a clockwise manner. And so that's why clocks were then later, when they were given gears and gears and, you know, cogs and such, they were designed to go clockwise because they mimic the sundial's direction. So it's just because of the rotation of the earth that we got clockwise. What if we run a different planet and it's spun the other direction? And so sundials move counterclockwise. There's no, there's no big importance there whatsoever. So again, well, okay, if it's arbitrary, why would we disagree with the culture here? Well, because of the way we orient the x, y plane, that is, we have our x-axis here and the positive direction is considered right and the y-axis, its positive direction is considered up. It turns out that angle measure will make a lot more sense moving counterclockwise than clockwise. So the way that we orient coordinates in the x, y plane is typically in the following manner. We draw a horizontal line and we denote this as the x-axis. We draw a vertical line and we don't know that as the y-axis. The intersection of the x and y-axis is commonly referred to as the origin as to, in a manner of saying that all points originate from the, this point, the intersection of these two things. So if you have a point in the plane, you can count how far along the x-axis is it, that gives you your x-coordinate and how far along the y-axis is it, that gives you your y-coordinate. That's how coordinates work in general, okay? So that gives us the standard coordinates of a point in the plane. We can also talk about the standard position of an angle because angle is determined by any three points in the plane, right? We just connect them with the rays. But as the measure of the angle doesn't change if we rotate the whole angle, something like this, the exact location of the points don't matter if we're just interested in the angle. So we can put an angle in so-called standard position if its initial side coincides with the positive x-axis, which is this part over here. The right side is considered the positive side. And then the terminal side, well, it just terminates wherever it wants to, okay? And so an angle in standard position would be something like this. Its initial side always coincides with the positive x-axis if you're in standard position. And then the terminal side is whatever it is. And so there's some point over here, x comma y on the terminal side. So if you give me a point other than the origin, that'll determine a unique angle in standard position. And so moving forward, we'll often do that. We'll describe angles via a point on the terminal side, a so-called terminal point that determines the angle, okay? So another thing we should mention about these coordinate systems, the idea of a quadrant. A quadrant, as the name suggests, is just one fourth of the plane. The first quadrant is going to be everything above the x-axis and to the right of the y-axis. And then going in a counterclockwise rotation here, the second quadrant is going to be above the x-axis to the left of the y-axis. Things in the third quadrant are exactly those which are below the x-axis and to the left of the y-axis. And then the fourth quadrant are all of those points which are below the x-axis and to the right of the y-axis here. And so we get these four quadrants. And so the angles that determine the four quadrants, the so-called quadrental angles, are going to be important to us. So the x-axis coincides with the angle zero degrees because the x-axis, if your initial side is the x-axis and your terminal side is the x-axis, then of course that's a null angle that's going to give the x-axis. That separates the first quadrant from the fourth quadrant. The first and second quadrant are separated by the angle of 90 degrees. The second and third quadrant are separated by the angle 180 degrees. Notice that 90 degrees is a right angle, 180 is a flat angle. The angle that separates the third and fourth is going to be the angle measure 270 degrees. And if we were to go one more, we're going to end up with 360 degrees. Well, 360 degrees and zero degrees actually represent the same angle in a manner of speaking. It's not the same angle measure, but it's the same angle. So what's the difference? Because in terms of an angle, an angle is determined by two rays.  That's what determines the angle. In terms of the measure, we're trying to think of how much space is spanned. Are we taking this angle right here? Are we taking this angle like right here, right, where you do a complete rotation plus a little bit? Or what about this angle right here where we do two complete rotations in a little bit, right? Those all in terms of their initial and terminal sides are the same, but the angle measures are different. This gives us an example of what we call coterminal angles, coterminal angles. Coterminal angles are going to be those angles which have the same initial side in the same terminal side, although their angle measure could be different. For example, zero and 360 degrees are coterminal angles, as observed right here. Let me give you another example. Take, for example, 60 degrees. A 60 degree angle is coterminal to 420 degrees. That is, if I put it in standard position with the positive x-axis, a 60 degree angle would look something like this. A 420 degree angle would be exactly this thing right here. In fact, we know that two angles are coterminal. If their angle measures, if you take the difference, is a multiple of 360. So notice that 420 minus 60 is 360. So that shows you that a 60 degree angle is coterminal to 420, because their difference is 306. That means they're only off by one complete rotation. It's also true that 780 is coterminal to 60. Notice that 780 minus 60 is 720. 720 is three times three, or excuse me, is two times 360.  You'll also notice that negative 300 degrees is coterminal to 60 degrees, because if I take negative 300 degrees and I subtract from it negative 60 degrees, you get negative 360 degrees, which is negative one times 360 degrees. So the difference there would be that starting at the initial side, you go clockwise 300 degrees until you get the terminal side. And so negative 300 degrees is likewise coterminal to 60 degrees. For the most part, there's no reason to distinguish between 908 degrees versus, say, 188 degrees. And there's really no reason to distinguish between negative 800 degrees with 280, because these are all coterminal with one another. There are a few exceptions to that, which I will point out along the way. But for the most part, there will be no difference in terms of measuring different coterminal angles. And so because of that, we will choose our angles to be between 360 degrees and zero degrees. It should also mention that sometimes we need an angle measure that's less than a single degree, because after all, the degree measure puts 360 degrees inside of a rotation. But what if we need something more refined than 360 degrees? In that situation, there's sort of two common ways people use to subdivide a degree. One approach is just to use sort of like a decimal portion, decimal portion of that. So for example, you might get something like 15.34 degrees. And so just to make sure 15 degrees plus 34 100th of a degree. And so sometimes to be more precise, you might need something like that. Another approach that was used more historically, it's not used as much nowadays, but still there are some that still use it is the idea of like a minute. That is a minute is 160th of a degree. And so you might say something like, Oh, I'm going to take 15 degrees and 20 minutes. You put a little apostrophe there to indicate a minute. But how do you what do you do if you get more precise than a minute? Well, one minute is 160th of the degree, right? So one minute equals 160th of a degree where here minutes are measuring, measuring angles not time. You can also subdivide a minute in the usual way that if you take a second, this is going to go 160th of a minute. In which case, then more precisely be like, Oh, 15 degrees, 20 minutes, 24 seconds. And so that gives you pretty some pretty good precision, right? You get up to including minutes and seconds, you can get up to one out of 300, the 3600th of a of a one degree, which is already pretty small itself. So there's very not often where you need to be more refined than minutes and seconds. And so that's a standard unit that is used over and over again. This is kind of similar to like, if you pull out a ruler or measuring tape, if you have like an inch, right, it might be subdivided into a half inch, and then like a quarter inch, and then like an eighth of an inch, and then you have like, what do I say, like a sixteenth of an inch, I keep on going or a 32nd of an inch, you don't typically see very many measuring tapes that have more than 30 seconds of an inch. That's pretty refined. There's not many measurements that require you to get smaller than a one 32nd of an inch. The same thing is also with degree measure. Now for this lecture series, I'm going to exclusively be using just decimal representations. So if we don't get a precise degree, we're just gonna, we'll just route, we'll just do a decimal. But be aware that in some angle measurements, they still use this idea of minutes and seconds.


Good evening. I'm Bob Baldacci and welcome to Baldacci on Business. This is our second show in a series that we're introducing to the viewers of CTN, Channel 5, called Pitch Me. And each month we're bringing to the studios and appearing before our very distinguished panelists, an entrepreneur who has a business idea, would like either to get consulting services or actual investment in their company. So this is our second show here in November and I'm very pleased to introduce our guest entrepreneur, Brian Crowder, of Life is Fluid. And Brian, we're really pleased to have you with us tonight. Thank you. And before you make your pitch, I'd like to introduce our panel. First, Mr. Hugh Stevens, Director of the Knowledge Transfer Alliance at the University of Maine. Hugh is an investor. He brings to the table a lot of resources from the University of Maine and is a former owner of a clothing manufacturing firm in the United States. Elizabeth Baldacci, my partner in Baldacci Group. A lot of experience, sales and marketing experience, 18 years. Elizabeth, thank you for being here this month. Again, we're very pleased to have with us our guest panelist, my good friend, Mr. F. Lee Bailey, who has an equally famous career in business as he does in the law. And Don Gooding, my good friend, Director of the Maine Center for Entrepreneurial Development, Vice Chair of the Maine Angels, and an investor as well. So panelists, thank you. Should be fun. Mr. Crowder. Thanks, all yours. Well, guys, as we see here, whether it be plain water, I emphasize plain or an enhanced beverage, we've got a problem with single-use bottles. In fact, two billion single-use bottles are thrown away in the United States alone every single month. Life is fluid, but it shouldn't be one single-use bottle at a time. So we've created a revolutionary new hydration system called Life is Fluid. And the idea behind this is quite simple. We intend to replace single-use bottles. And there's some other ideas that I will approach, including all U.S. manufacturing. And in case now, in the case now actually Maine, which Maine jobs, Maine production. The product itself is about three and a half years in the making. And a lot of that has been put into the R&D, different patents. We have some patents pending right now. We also have a flask skin. And the production in the United States is really what kept us going and kept us sort of moving forward and yet took some time. And now we have the Life is Fluid life flask. And the flask skin, which is, as you'll see in a second, is the sort of the branding mechanism, but also provides some utility. As I mentioned, the patent doesn't tail the twin chambers. You see this is a hydration system, more so than just a plain bottle. And there's two chambers. And within the second chamber, which is the real key, is a micro cartridge, which is filtration and enhancement. This particular one is just an NSF-42, so it's like a Brita pitcher on the go. But we can also add enhancement. I won't mention any brand names, but we can add a flavor. We can add vitamins as we move through the R&D system. One of the most interesting things about this product is the fact that it's essentially a living product. We mentioned before that we have a problem with plastic ball use, whether they end up in ocean gyres or landfills. This is a living product. It means it starts with 25 percent recycled material. It's made at main manufacturing in Sanford. And we start 25 percent recycled HDP, too, same as your milk jug. So it's the BPA-free, everything that you guys have heard about. The 75 percent virgin is then added in to make the whole product. If you buy one of these, or when you buy one of these, I should say, for the life of the product, we will, for the cost of shipping and handling, send you back a new flask, the Life Flask. And a return envelope, postage paid, to main manufacturing in Sanford. And they will recycle it on site, thus becomes 25 percent of the new one, and it becomes a living product. It really is designed, as you'll see in some of our corporate social responsibility strategies, it's been designed to get rid of single-use bottles, as well as produce main jobs and U.S. jobs in general. But recently I started working with Allen Manufacturing up in Lewiston. They're actually working on, you see here, we have two products, two different skins, essentially, and we're trying to decide which ones. If you mind, let's pass these out. You'll notice just slight differences. As part of the product R&D that we've been in for a few years, this flask in, which can now be made up in Lewiston, and they're doing a time study on both of those right now, as we speak. The entire product then will move from main manufacturing in Sanford through the material, which is expanded rubber products, also in Sanford, through Lewiston, which is Allen Manufacturing. They'll make and distribute from there. So it's something that, like I said, this is in the last six months, but we've really worked hard to make that happen. As you see behind me, we talked a little bit about the product and the elimination of the demand for single-use products. But the reality of the situation is we've also created a company and a brand behind Life is Fluid and the Life Flask that believes in the concept of redistributive consumerism. My own personal background is Wall Street Economics, Economics Degree from Michigan State, and then an MBA with an economics major at Berkeley. And when I started this company, the idea was, how can I take a for-profit mentality, make as much money as possible, which no doubt I'm a capitalist. For those of you who are maybe thinking about down-the-road investing, it's a for-profit company. We've got 80% gross margins. We've worked hard to make sure our fixed costs are low and our variable costs are, you know, like I said, 80% gross. So we've got that really well under control. But when we sell one of these, we've created a program called You Buy One We Give One, which is similar to Tom's shoes. For everyone we sell, we're going to donate one to Just Causes. This range is from, I've already done a sponsorship with Special Olympics. We've also obviously with the filtration, as the filtration gets better, and we start moving into third-world applications. If you can also donate to the third-world applications. Our programs, or more importantly, how we're going to sell these, because as you see, we've created, we've sort of bitten off a lot in the last three years. We've got the USA main production now. We've got the You Won't Give One program.  Then the more money we make, the more money we can give away. It really is the DNA and the background of the life is fluid brand and the life is fluid concept. We're going to do this through two main programs that both take advantage of direct marketing. The advent of, or the invention, if you will, of social media, social marketing, and QR coding. Different ways to reach your consumer directly means more profit for the company. It means shipping and handling. It's the Amazon model. We've all seen what's happening with Best Buy in Amazon. It's here to stay. In fact, in the month of June 2012, 25 million different people downloaded QR codes. This QR code, if I had a phone, which I don't because I have to be turned off, I could scan this code right now. It would take you directly to my site where you could purchase one and move on. We've got a wave creator program, which is essentially a multi-level marketing or a way to get our loyal active lifestyle brand people involved. If you'll notice, I'm going to pass these out as well, if you become a wave creator, not only do you receive free shipping, but you also receive 100 of these cards that are replaceable, which have a QR code that is specific to that wave creator. If you're on a share lift, we're going to actively attack in this first round the ski ride category. This is very active. As you can see, as the panelists can see, it's lightweight. It's something that, if you're skiing, you can fall on it. It's perfect for skiing. Matter of fact, I've been using it for a couple of years doing the R&D process. But if you're on a share lift and you take a nice swig of hydration or hot chocolate, the person saying next to you says, what is that? And believe me, they do. You can see on the top of the cap, there's a QR code. You may also have this code or your specific code. This is LIF 001 for life as fluid wave creator number one. When that person goes to the site directly from that code, you receive 10%. And the back end of this site has been engineered to handle all of this. It's kind of got my Wall Street background on it, if you will, because that is going to be a lot of tracking. But the way the back end systems in the web work these days, it works very, very well. Retail is a question that always comes up. We have talked to different retailers. We've had interest in retailers. But as part of the whole concept behind the Amazon Best Buy, what retailers are asking for in terms of 52%, 55% of gross in order to charge you another dollar for charged backs, and basically for shelf space, which in my estimation is really sort of overvalued these days. We've created a pipeline program. And the pipeline program is essentially a one by two kiosk, has a couple of sample products. The retailer has a code just like the ones on the wave creator that we mentioned. That retailer then gets whatever they negotiate, let's say 30 to 40%. LIF as fluid maintains shipping and handling. We maintain control of our brand. The space in a retailer, many of our retailers because of our active lifestyle segment, are smaller shops. They're Dave's Tahoe ski shop, Arlberg. As we get into these shops, these guys that we've talked to so far, which we have, have already liked the idea of a poster. Maybe a couple of these cards, they throw them in the bag. And all somebody has to do is say, oh, where can I get one of those? And in the case of the chairlift, by the way, the way that the UPS works these days, by the time you scan that on a chairlift and got home on Monday or Tuesday, your life flask is waiting for you there. So that's the general overview. And as I apologize, eight minutes for three and a half years, especially with somebody as verbose as me, isn't so easy. But one of the things I want to make sure I touched on before we move to the next level was the products in R&D. One of the great things about the hydration system is if you'll notice there's a, the handle, if you will, which is one of the patents, has a place for a clip. So we've got a bicycle clip. We call the flask clip in production right now in R&D at main manufacturing. And also we're calling the flask quiver, which is essentially a ski patrol type harness where this could sit right on your chest. And that just adds, obviously, a whole another level of marketing ability as we move into that active category and active lifestyle. Excellent. Brian, thank you. Panel, fire away. Questions. How many of these have you sold so far? I haven't sold. Well, I've technically sold as, as, as Bob mentioned, when we first got started, we started with a different product. We sold about a hundred of them. I realized, you know what, this is something that I need to take larger. That's when I started working on my own patents. But we are pre-revenue. The site and everything is up and ready to go. When I get the time study, I should find out from Allen Manufacturing, we're about three weeks off from the first order. And I already have 5,000 of the units with the filtration sitting at main manufacturing ready to go. So we're pre-rev, but we're ready to go once the flask is in our hands. So, so by the end of the first quarter of 2013, how many do you hope to have sold? We have, we went through in the pamphlet that you looked at earlier. We have the sports and sports, SGMA is called the Marketing Association, which breaks down users of different active categories. For instance, ski ride is the first one we're going to. We've broken down that market, core enthusiasts only, people at ski and ride more than 50 times a year. We took some measures of that, right? 6% of this captures some reasonable conservative assumptions. We're expecting approximately 25,000 unit sales in ski ride category by the end of ski ride, which fits in, I think, with your question, which is right around April, May. We'll pick up some extra. We'll pick up mountain bikers, hikers. I moved out here from California. As, as anybody knows, we call it a ski ride category. Out there you can golf in the middle of January, February as well. So other categories will pick up extras, but we're going to really focus on earning and owning that active category, the influencers within the ski ride category. So to answer your question, 23 to 25,000 just in the ski ride category. Mr. Bailey. The wave creator phase sounds a lot like a multi-level marketing system. Amway stuff. Excuse me? Amway. Amway stuff. Good example. Correct. It is. I mean, I think that's a great way to put it. Amway obviously has a name sometimes, depending on how you feel now. I'm from Michigan, so I know a lot of people from Grand Rapids, and they may feel differently about Amway than most of us do, but to be quite honest, the multi-level marketing is a very successful sales marketing technique. It can be overrun. It can become, if I dare say, cult-like. In this particular case, because we are going after an active lifestyle market, and we do, we believe, have a wonderful branding from the name, Life is Fluid, the Life Flask, everything that goes around it, the you buy one, we give one. We're sort of trying to build a base that, if we do it right, could become cult-like. And with the margins that we've got, we can step through sales. We don't have to step into this product now after three years in the partnerships that we've made with main manufacturing, et cetera. We can sell a thousand a month for two or three months, build a base, go through the outside magazine channel, go through some of our connections, and really start to build that base up. We don't have to sell 10,000 in the first month, 50,000. Just an observation, but I see that you have arranged for bikers to be able to carry one with them. What about people who own cars? Well, it actually fits. There's a lot of different ways for it to fit in the car. The best way for it to fit in the car is actually it sits right down to your right side between the console and your seat. We also, because of the Flask Clip, and again, the easiest way to describe it is if I can grab one real quickly from one of you, Joe. The design here has created what we call a Flask Clip, and the Clip is very, very lightweight. If you notice here, it's hard to read, but it's under 25 grams for a stainless steel, which is very inexpensive to make. That's a very lightweight Clip. It just sits in like this. So what we have on a bicycle is it clips in like this. This Clip system can be put anywhere. So the next, one of the product R&Ds around this Flask Clip and this design is to put it in your car, or maybe it sits up just like a regular bottle holder, and it just sits in this Clip. And that Clip is inexpensive to make. I mean, where we price that out already on an R&D basis with main manufacturing, made out of carbon, made out of stainless steel, and made out of an ABS plastic, like a high-end canoe, we have huge margins in that as well, which may lead me to the money aspect. Well, Elizabeth. I just want to talk about the practicality of the use of this. So you buy the container, and I know you have some patents pending I'll talk about in a second, but so I have it, there's Nike has them, everybody has these containers. When I'm driving in my car or I'm on a mountain, I think people get these single use bottles because they're easy to get. They're inner refrit. It's an easy source of the product. It's not, you know, it may not be the right thing to do, but you can grab four of them and get out. Where do you get your water from? Well, that's the beautiful thing about the filtration system. So, and again, the way this works is there's two chambers, right? You can mix, you can have two different fluids if you want, but the reality is tap water goes in here, and I use it for my children. Any tap water? Any tap or a creek. I could dump this into a creek. I've used a circle of golfing and dip it into the creek. When this caps on, you squeeze it, and it goes up through the filter, and what's in here becomes filtered or enhanced. Okay. So to answer your question, part of the reason I came with is I have a bunch of small children, and when you travel, we used to drive to Tahoe three and a half hours. There's a McDonald's. Well, trying to convince a six-year-old to drink McDonald's water out of the tap in the bathroom is not that easy. So instead you end up buying a $2 bottle of water, which we've already discussed some of the evils around that, despite the fact, despite the fact it's $2. With a product like this, a mother, a father, a child, wherever the source is, they can flavor it, they can enhance it, they can filter it. It now becomes, like I said, a bit of pitcher on the go. And that's really the value add. It's one of the best value add. I mean, the other value add that we didn't get into, but it is an active lifestyle. On a triathlete basis, this is a measured amount. So as you get more and more what I call the aspirational athlete up to the more accomplished athlete, knowing exactly what your intake is of your chosen beverage is very important as well. It's something that in our R&D phase over the last couple of years has really come back well. And do you have any disclaimers or legal protections against if somebody does take basically non-potable water, that there's some signs in restaurants and hotels, and they use this and they get sick? Standard. I mean, we've got, as part of our sources and uses, we've got corporate counsel out of Boston. They've worked through some of those. But the good news is both the same with the NSF 42, which is the, it's just basically changing the aesthetics of water. It's very straightforward, which you explain. And then with regard to that, yeah, your standard sort of look, if you drink this and you don't have the filter in, you could get sick. The good news is because it's ACP2, because of all the hub, you know, all the hub hub about BPA and everything that hit the last few years, ACP number two, which is water, which is your milk jugs, et cetera, has been, has pretty much been covered. So I think everyone's comfortable with that. Tell me a little bit about your company as it is now, structurally. Sure. Well, like I said, I'm the CEO slash chief wave creator. And I've been at this obviously from the beginning. I've worked with main manufacturing now on the manufacturing side for about three and a half years. So they're a partner, although it's through supply agreements. We've got the guys that I mentioned before on the production side. As far as the company itself, I work with Jim McGinley. He's with the Fin studio, their design and branding, and also a main company headquarter here in Falmouth. We work with KL Gates on the patent. We worked with University of Maine, with Woody, initially about three years ago, and he gave us a stack of information about potential patents and what to get in trouble with. But our patents pending are through KL Gates out of Boston. And then we work with Perry, Crumsick and Jack out of Boston on the corporate side. So we've got what I would call a production team in line. We've got two different people competing for the distribution. Shipwright Solutions out of Portland, and also SPM Distribution who handles Toms of Maine. They're in Sanford. And of course, Allen Manufacturing, as you would expect out of anybody trying to gain more business. They would like to do our distribution. That makes a lot of sense because shipping a bunch of these lightweight containers to Lewiston, and then having the final step of the process be the one that kicks it out is a really good idea. Okay, other questions? Comments from critique. From our panelists. I have to say that spending three and a half years to get to this point without having sold more than 100 is kind of unsettling for me. Because you created a very complex set of products and marketing and programs and all of this stuff. And I guess I'd like to hear what kind of validation you have that any of this makes sense at all. Because, as Elizabeth was saying, it's an extremely competitive market that you're going into. Absolutely. And it gets back to two things. It's a great question. And like I said, believe me, no one feels this three and a half you're paying more than me. But a lot of that had to do with trying to maintain U.S. production. I could tell you it's some horror stories, but the reality is we got here. On the branding and the personnel, on the decision of what it looks like, how does it work and how well it works. Two years ago, I started in earnest getting them out in little groups and even though I haven't sold any, I've donated in the neighborhood of 700 to different events, including ski events, golf events, biking events, even pre-clip because it does fit sideways. To sort of test the idea, do people like to squeeze? What's the shape? Does it feel good? We've made changes throughout, including adding a filtration. But in essence, to answer your question now, we're counting on the fact that the team I've put together, including Jim and his team at the Fin and everyone else, significant backgrounds and activities such that we're attacking, ski ride, et cetera, in addition to mountain ice or race mountain bikes. We've tested it extensively within our own group. So it's that we're pretty secure that the first one that goes out isn't going to be perfect, but it's going to be really close. And because of our gross margins and our fixed cost being so low, we will be able to change on the fly and make adjustments on the fly and figure out exactly what part of a market. For instance, golfers may find that they like a part of this much more so than a ski ride category. We're a camper who's just going to use it for the filtration. As we move through the process, we've kept our fixed costs so low that we'll be able to direct market, affiliate market, event market to those people at relatively light costs on a fixed basis. Comments, questions. What are your needs? Well, we need money. How much money? We're asking for $250, up to $250,000 on a royalty basis. With my finance background, I could talk equity and equity valuations until I'm certainly tired of hearing myself talk. But the reality is the margin basis that we've set up, which like I said is in the neighborhood of, it's 80%, which works out to be about $24 each after sales, marketing, et cetera, allows us to do what I like to refer to as our sort of PO program, right? If I went into LOB and I sold $10,000 on a pre-order, that sounds really good, particularly in a panel like this. Oh, you've already sold $10,000 to LOB. The problem is I sold $10,000 at $14. I had to produce $10,000. I had to ship $10,000 to them, et cetera, et cetera.  And we'll say, we'll give you in this case, we're offering $6 a unit until you get to 150% return.  So you become a top-line royalty expense item. Obviously, you want to make sure we keep playing money around for marketing and everything else to make your sale. But that's what we're offering. Up to $250,000 at $6 a unit. I'm open to other conversations. It's just as very difficult, as you can well imagine, to really value a company like this because we have spent a lot of money. We do have a lot of partners that have spent a lot of money. So just valuing it on a pre-revved basis is very difficult. What is your suggested retail price? $29 for the full set, which fits pretty nicely actually into the market of, for instance, a camelback is $21, single chamber, no skin, neoprene, carabiner. So from a pricing standpoint, that's the other thing that we've been able to do over the last couple of years. We've paid for different marketing studies and different things like that over the couple of years to say, what do you think about $29? What do you think about $7 shipping and handling? And it fits pretty nicely in, filtration, for instance, is $40 enough. And recently they came up with BobaWater and some others that are $9 or $10. That by the way, BobaWater sold 400,000 units in the first six months. It was $10. It doesn't work very well. It's made out of PET. You can't leave it in your car, but it's another issue. So $29 plus $7 shipping and handling. Can you give one away for the $29 or is that a different price? Excuse me? Though buy one, give one away? Is that $29 and you give one away? We give one away. Okay. And what ends up happening, as I said before, the actual cost to produce these is actually, you know, we've got it with a very good manufacturer, from a competitive standpoint, by the way, not much different than what I could have these produced overseas now with main manufacturing on a per unit basis. The flask in that you see, and you've seen a couple here actually, will be, you know, just a basic skin with a basic information that this is that you want to give, we give one. Panel, we only have just a minute or two left. All production efficiencies are thoroughly and exhaustively been researched in your... They absolutely have. The only thing we're dealing with now is breakpoints. So I've worked with these guys. Okay. Any interest in working with Brian from the panel? Feedback, quick. We only have just about a minute left. Yes. When the Advanced Manufacturing Center was in conjunction with my program, KTA would be very interested in helping in assisting. Thank you. Thank you. Elizabeth? I just don't know enough about the space. I think your marketing and branding is brilliant. I love it. I think you've done a really good job with it. So I'm going to pass. Lee? If they can master you for multi-level selling, call me. Okay. And Dawn? I'm skeptical, but I've been proven wrong many times. Okay. All right. Well, we have three minutes left. Reaction, obviously you've got strong interest from the University of Maine, and Hugh does bring a lot to the table. Is that something that you would like to... Absolutely. I mean, one of the things as I hope to hit on earlier was the whole Maine concept, moving out here from San Francisco, moving my family here, working with the University and working with Hugh would be fantastic. And I agree with Beth. I mean, some of the problem of doing this for three and a half years is we need money, and we do need expertise, but we've got a team that we've been working with for a long time. I'm glad to hear that you like our marketing. I like a lot. It's clearly taken a while, and we're finally happy with it. And if anybody comes after me, I'm definitely going to give you a call. And Dawn and I have known each other quite a while, and I think what he's doing at Top Gun and everything he's doing is fantastic on the startup front. Yeah, and if I can't... I just want to advise one thing, and I've known Don a while now since the Maine Angels. He's brought new energy to that program and to the Maine Angels, and I think to the extent that you can kind of work with him and work with his resources, including the Top Gun program. Now, I think that would be a great way to go. I like... I think the world of you, and I love your concept. And again, I really appreciate you being on the show, Brian. Thank you very much, guys. Thanks for your time. And panelists, thank you very much. So we conclude our November show, and again, I want to thank the viewers who are watching tonight for their time and attention. I hope you've enjoyed it, and I look forward to you folks joining us for our next season. If you have any questions or comments, please feel free to give us a call. You have our website information on the show or call the TV station. So again, thank you very much. We really appreciate it. Thank you. Brian, thanks.