So, we've seen now this equation that tells us the slope of one of these coexistence lines on a phase diagram. That equation is called the Clapeyron equation and we can determine that slope if we know the enthalpy and the change in volume and the temperature of a phase transition. So we'll see how that works with an example. We'll take a substance that we're fairly familiar with, water. So if I tell you that the heat of vaporization, the molar enthalpy of vaporization, or actually not vaporization, we'll do fusion. So the molar enthalpy of fusion, when I melt solid water to make it liquid, is 6 kJ per mole. I can also tell you the molar volume of the liquid, the molar volume of the solid. Probably the easiest way to think about those is we could look up pretty easily the density of the liquid. The density of liquid is pretty close to 1 gram per cubic centimeter. Look up the density of solid, the density of ice. And then convert those densities to molar volumes relatively easily. I'll just give you the molar volumes. One mole of water, 18 grams of water, takes up about 18 milliliters. So that molar volume is 18 and if I need a few more sig figs, that turns out to be 18.0124 cubic centimeters is the volume occupied by one mole of liquid water at its freezing point. The molar volume of ice, we know that ice is less dense than water. Ice cubes float in water, so the molar volume is going to be a little bit larger. If we look at what the number actually is, the density of ice at the freezing point is a little over 19 and a half cubic centimeters per mole. So that same mole of water expands and takes up a slightly larger volume when it freezes. That's enough information combined with the fact that we already know the freezing point for us to calculate the slope of that solid liquid coexistence line using the Clapperon equation. So if that's our question, what is the slope dp dt for the solid liquid coexistence line? Clapperon equation tells us how to calculate that. Enthalpy of fusion, 6.01 kilojoules per mole or 6,010 joules per mole. The temperature I need is the temperature of fusion, the temperature of the melting point. So that's 273.15 Kelvin, freezing point of water. And then we need the change in the molar volume for fusion. So remember the fusion process, when we say enthalpy of fusion, temperature of fusion, molar volume change upon fusion, the fusion process is melting, so solid becoming liquid. So it requires an input of 6 kilojoules per mole to convert lower energy, lower enthalpy solid into higher enthalpy liquid. So we have to make sure we're doing this in the right order. The change in volume upon fusion is the change when we go from solid to liquid. So the delta V is volume of the liquid, 18 cubic centimeters per mole, minus the molar volume of the solid, 19 and change, cubic centimeters per mole. And that would be that, except we need to be a little bit careful about units. Moles are fine, the moles in the denominator of the denominator, cancel moles in the denominator of the numerator. But we're left with joules and Kelvin and cubic centimeters, none of which seem like they'll cancel each other. We can, however, do some unit conversion. So let's multiply this whole thing by, a joule is not equivalent to a cubic centimeter, but a joule is equivalent to a liter, some number of liter atmospheres. Remember, energy can be expressed in units of joules or as a pressure volume product as liter atmospheres. So recalling the values of the gas constant, the gas constant is either 8.314 in units of joules per mole Kelvin, or .08206 in units of liter atmospheres per mole Kelvin. So that unit conversion is going to allow us to convert joules to liter atmospheres. And now at least we've got a volume here, and if I write down volume as liters in the denominator and cubic centimeters in the numerator, these liters will cancel, cubic centimeters will cancel, and I'm in good shape, and I need to know that there's a thousand cubic centimeters in one liter. So then by the time we're done those unit conversions, we've got an answer that's going to be in units of atmospheres in the numerator, and I've still got a Kelvin that hasn't canceled in the denominator. So that's what I expect. That's a ratio of a pressure to a temperature should come out in something like atmospheres per Kelvin. The units have worked out fine. If I do the arithmetic of calculating these numbers, what I'll find is that that works out to negative 133. Notice the negative sign came because I took a small volume of the liquid, 18, and I subtracted a larger volume for the solid, 19. So this quantity in parentheses is a negative number, which is where that negative sign came from. So what we've learned from the Clapperon equation is the slope of the fusion coexistence curve, the slope of the solid-liquid coexistence line on a pressure-temperature phase diagram, solid-liquid gas. The line we're talking about is the solid-liquid coexistence line. The slope of that line, I've drawn it leaning backwards because Clapperon told us that the slope of that line is negative 133 atmospheres per Kelvin. When I change the temperature by positive one Kelvin, the pressure drops by 133 atmospheres. So that's quite a steep curve. It's leaning backwards, but it's pointing upwards quite close to vertical. So that's one thing that this Clapperon equation tells us. We can calculate numerically what the slopes of these lines are. So near the normal freezing point of water, the slope of that line is this particular numerical value. The other thing it tells us that's interesting is in fact the sign of this equation. The one we've just calculated, the solid-liquid coexistence line, we now know slopes backwards when we're talking about water. On the phase diagram for water, that solid-liquid coexistence line has a negative slope. For other substances, it won't work that way. So for most other substances, it worked this way in particular, as I pointed out, because the volume of the liquid is a smaller number than the volume of the solid. So when the volume of the liquid is less than the volume of the solid like it is for water, or conversely, if the density of the liquid is greater than the density of the solid, then the curve will look this way. If those things are reversed as they are for most substances, if we have a substance where the molar volume of the liquid is larger than that of the solid, or the density of the liquid is less than that of the solid, that would be a substance whose ice cubes sink rather than float. So a substance whose denser in the solid phase than the liquid phase, so the solid form would sink in its own liquid. The phase diagram for that substance is still going to have a solid and a liquid and a gas. But that phase diagram is going to have a solid liquid coexistence curve, a melting line that leans forward. It's going to have a positive slope. Most likely it'll still be quite a steep curve, quite a steep slope, because the volumes won't be all that different from one another, but sometimes it'll slope negatively, sometimes it'll slope positively depending on the relative densities or the relative molar volumes of the solid and liquid phase. The last thing we can learn from this type of calculation, in addition to the slope of this curve, we can think about what that slope means physically. It's perhaps a little difficult to think about the slope in this direction. If I say, if I were to change the melting point of water by a Kelvin, that's going to change its vapor pressure, not its vapor pressure, sorry, its coexistence pressure by 133 atmospheres, that's kind of difficult to imagine. It's a little bit easier to understand what that means if I do it the other way around. If I ask, by exerting pressure on the substance, how much do I change its melting point? So that's just the reciprocal. I've turned this derivative upside down. So that's going to be 1 over negative 133 atmospheres per Kelvin. And that's going to work out to still negative, but now a fairly small number instead of a large number. And that one's easier to think about physically. So the change in the temperature of fusion, the change in the melting point is negative, let's call that negative a little less than a hundredth of a Kelvin per atmosphere. If I change the pressure by one atmosphere, I'm going to change the melting point by negative 0.00754 Kelvin. So instead of melting ice at ambient pressure, one atmosphere, if I were to melt that same ice by exerting two atmospheres of pressure on it or ten atmospheres of pressure on it, if I increase the pressure by a few atmospheres, I'm going to decrease, because of the negative sign, I'm going to decrease the melting point by a few hundredths of a Kelvin. So that's instead of melting at one atmosphere and the normal melting point, if I increase the pressure to a larger pressure, the melting point will have shifted slightly negative. It'll be a few hundredths of a degree less than 273 Kelvin. So we can decrease the melting point of ice by exerting pressure on it. If it weren't water, but some other substance, I could increase the melting point of that substance by exerting higher pressures on it. So those give you some examples of how to use the Clapeyron equation as well as the significance of the results we achieve by using it.


Good afternoon everyone, and thank you so much for joining us two weeks ago. Governor Wolf gave his 2017-2018 budget address in front of the state legislature. He has third budget address as governor. Today we are so pleased to be joined by Governor Wolf, who will be here to answer your questions from Facebook on his plan for Pennsylvania. Welcome. Thanks Megan. And I just want to give you an opportunity before we get to all the great questions we've gotten on Facebook over the weekend. Just kind of walk us through what this third budget is all about. It's a little bit different than the past two. It is. It's a different approach, I think, to budgeting. In the past we've been given a choice between the paying more or getting less. And what I'm trying to do here is say, I'm not going to raise broad-based. I'm not proposing raising broad-based taxes. And what I'm proposing still is an increase in things that people of Pennsylvania care about, like education, like the opioid epidemic. Like seniors, like jobs. Government ought to be helping provide services surrounding those areas. Those are things that are of interest to people of Pennsylvania. I'm doing that in my budget. That's what I proposed without raising taxes. And just a reminder, these questions have been submitted by Pennsylvania citizens on Governor Wolf's Facebook page, where we are holding a live town hall right now. And we have a lot of questions to get to. We're going to start with sort of the news out of Washington these days, which is the Affordable Care Act being in flux. And so Julie would like to know, how will block grants from the federal government affect your budget? Julie, that's a great question. I don't think anybody knows at this point. Here's what we do know. Over the last two years, 700,000 Pennsylvanians have gotten health insurance. Didn't have it before. Over 120,000 Pennsylvanians are getting treatment that can now have treatment and pay for it for opioid, for substance use disorder. Those are things that Pennsylvanians now have. And we don't know what's going to happen in Washington, but whatever happens, I think the people in Washington have to recognize that there will be severe consequences if any of those 700,000 people find themselves without health insurance or any of those 120,000 people find themselves without the ability to get treatment for opioid substance use disorder. All right, let's move on to property tax. We got a lot of questions about House Bill 76 and just a lot of folks weighing in and saying they can no longer afford the massive school property tax. Wendy from Monroe County says I cannot retire in Pennsylvania because I won't have enough to pay my school tax bill and I don't want to move, but I will have no choice, she says. So can you address sort of what your plan or what your hope for property tax reform would be in Pennsylvania? Yeah, and Wendy's point is absolutely right that there are people who are being forced out of their homes because of increasing property taxes to fund public education. I ran on the issue of property tax relief and that's up to and including property tax elimination for the purpose of funding public education. We simply cannot have a system that forces people out of their homes as our main source of funding for the education for our children. We need that education. We need our children to get that education. We can't do it at the expense of our citizens who are being forced out of their homes. And you've talked to Republicans and Democrats across the aisle? I have talked to Republicans and Democrats of all political stripes and I think there is real appetite for real property tax reform. I think the discussion debate has to be around how we do that, how we implement it, the transition. But I think people in this building get it. We need to have property tax reform. Let's go back to the federal government. Eric asks, what are you planning to do about Pennsylvania driver's license that are not going to be valid identification for air travel inside the US that's upon us? Yeah, he's Eric's talking about the real ID program and I actually, Act 38 was established many years ago before I was governor. But the consequences are coming due now. So I called to then Secretary of Homeland Security, Jay Johnson, and they asked for an extension. And he agreed to give the extension until June as long as I could get a letter signed by the majority leaders of the Senate and House and I would sign it saying we will work very hard to get the repeal of Act 38 and that, I think, starts in the Senate and they are bound and determined to do that. If we get that repeal by June, then we have until I think the middle of October of 2020 to actually implement the program and that will give us plenty of time to implement it. So the hope is that we actually can use our driver's licenses to get into buildings that we can use our driver's licenses to actually show our identification at airports. And I know PennDOT is, you know, at the front of this and they will keep Pennsylvanians informed of the different changes that we come about. Yeah, I think the legislature is working on this and the hope is that we can actually fix this. Let's talk about unemployment compensation and from Harleysville says, what is being done about PAUC? This is not a giveaway we pay in insurance premium and so do employers. And there's no ability to get service when you need it.  Yeah, that's that's a fair point. We actually just to give you some history, we actually reformed the unemployment compensation system for seasonal workers, for cyclical workers so that we added, I think, 40 to 50,000 workers who were now eligible for benefits. And we did it without jeopardizing the solvency of the system. One of the details of that reform, which everybody, Republicans and Democrats, senators and representatives signed on to, was to keep the call centers open for an extra another year and keep funding them out of the unemployment compensation reserve fund and the trust fund. And and when we actually agreed on how we would extend these benefits, a small part of that was to authorize the governor and the Labor and Industry Department to use money in the trust fund to keep the call centers open that passed a house overwhelmingly. Republicans and Democrats got to the Senate and they decided not to even vote on it. So I'm ready to to to sign that as soon as it gets to my my desk. In fact, I will drive in my Jeep anywhere from anywhere on vacation, a holiday weekend, whatever, to get here to sign that because that's all it takes. It's ready to go. And and the bill is is is ready. So it's it is a sad state of affairs that that the bill is there. The reform is there. But because the money was not authorized to actually make the call centers function as they were supposed to function, there are people who have to wait long, long hours to to get the benefits they deserve. Just for anyone who's joining us, we are live on Facebook right now with Governor Wolf, and we're answering a whole gamut of Facebook questions that are coming in from from your viewers on Facebook. So let's transition a little bit to arguably the most important issue of your administration, which is education. Robert from Erie says, I'd like to hear how you are addressing the per pupil discrepancy across Pennsylvania, where some school districts receive. He says almost three times as much per pupil spending than on other schools. What do you say to that? That's a great question. We we are trying to and I've worked with both sides of the aisle to produce a fair funding formula. The fair funding formula is based on it. Distributes funds based on the poverty rate, the what is it, the number of children who speak non English language, the tax base of the of the area, the number of children enrolled in charter schools takes all those things into account to give a certain percent number of dollars. Now, the fair funding formula only relates to new dollars going into education since I became governor. So I've put now four hundred, five hundred million dollars new new money in proposing another two hundred million dollars this year. So it would it would apply to to those new new dollars. But it would try to to make sure that we're giving the areas of greatest need the most money and transitioning again to rural health. Emily from Lancaster says the U.S. is facing a looming shortage of physicians by twenty twenty five. There will be a deficit of sixty thousand to ninety thousand providers. So what is Pennsylvania doing to ensure an adequate supply of physicians in the future? It's a great question that the I think one of the problems we face is is not just physicians in general, but physicians in certain specialties like family practitioners, primary care specialists, or in short supply. And part of that has to do with, I think, the nature of tuition rates in medical schools and the state needs to do something there, I think. But we're also working to make sure that that rural there's a geographic spread. Some of the doctors, the shortages are most evident in our rural areas. So we need to make sure that that we do a good job of supporting rural health centers. The Department of Health and the Department of Human Services have both been working with the. Health and Human Services, I think, at the federal level to to secure some funding to help us support rural health centers. The again, the Affordable Care Act that gives Medicaid to people, in many cases, who use rural health centers to actually reimburse those rural health centers for the health care. So there are a lot of things that we're trying to do, but it's it's a physician shortage issue that's looming. There's also the issue of geography. There are certain areas where that shortage is most acute. I think they're certain, especially. So we're trying to do work with all those things to try to address that issue. Marijuana, Liz from Erdeneheim, I hope I'm saying that right, says, is there any way Pennsylvania can follow Colorado's example and gain their amazing budget surplus over a quarter billion dollars by legalizing marijuana? Yeah, here's what we're doing. We've legalized medical marijuana. I ran on the legalization of medical marijuana and on the decriminalization of recreational marijuana. And I think that that both of those things will the first will produce some revenue. The second will reduce the expenditures on things like prisons and our legal system. I think personally that we ought to wait to see what happens in Colorado and Oregon and Washington to see how that that actually works out before we we move beyond what I've said I wanted to do and what we've already much of which we've already done. On to sort of back to your budget, this year's budget. David from Frizzleburg says, what what hope what do you hope to accomplish? He says, by raising minimum wage, knowing the effects it will have on the economy and that however high you raise it, that will always be the wage then. Right. I think what you're trying to do with the minimum wages is adjust it to a point that actually helps the economy, not hurts it. Obviously, and his point is to be raised it too high. It's not not a good thing, but raising it to the right amount, whatever that is, I'm proposing $12 an hour actually puts in this case, depending on your estimate, maybe to even more two billion dollars more into the economy. And so that's two billion more dollars that people have to spend on things like pizza and Big Macs and and food in the grocery store, clothing or items in the in the dollar store. These are things that that actually add to the the economic vitality of Pennsylvania and in terms of taxation, should be worth anywhere from additional 60 to 100 million dollars between personal income tax and the sales tax. So the economic activity is is what you're trying to you also should save money with a minimum wage. Right now, you can work full time minimum wage in Pennsylvania, $7.25 an hour at that rate. You work 2000 hours a year and you're still going to qualify for things like food stamps or the Children's Health Insurance Program. Both of those things cost taxpayers money. So a minimum wage to a reasonable level would actually increase economic activity and save the government and all of us taxpayers from having to spend money on social services. And that's something you've been behind since day one. And last year you traveled the state and you met with a lot of business owners. And it seemed like you got a good reception. I did. Well, and also I was a business owner. I paid well above the minimum wage and minimum wage was not something that that would be a problem, I think, for for business owners. This is something that actually on the whole in the aggregate would actually increase demand, which would be a good thing for business. Yeah, there's a lot of ways to look at it. Yeah, absolutely. Let's go to Sheila's question and I'm going to read it verbatim. She says, Dear Governor Wolf, thank you for voicing your support for better funding for adults with disabilities. I would like suggestions now from you on how families of the disabled can help make the legislature make the changes you've asked for. Well, thank you very much for that. I think the best thing is to contact your state senator, your state representative and say, I really support increasing funding for adults with disabilities, and I hope that that you will support the governor in making these increases come true. All right. OK, pensions. You ready for that? OK, Claudia doesn't give a location. She asks, What are you going to do about the crippling pension program? And we heard this from a lot of our viewers on Facebook. And she says, pensions are killing us in taxes. State employees just need to switch to a 401k, just like the rest of us. That's a good suggestion. I think state employees are willing to to go to a defined contribution plan. That's what the 401k is. The House Democrats and Senate actually passed a bill that would be a hybrid that would have a combination of both defined benefit and defined contribution. So to the Senate, they were not able to reconcile and get to a point that both of them could agree on. But I'm I'm ready to to sign a pension bill that does three things. It reduces the risk, reduces the expense for the commonwealth and is fair to employees. I think those are things that that we need. And I think she's right that that we can do this with a defined contribution plan. We just need to do it the right way. Let me just point out that the reason, the big reason we have this deficit is only partially because of the defined benefit design of the plan. The big reason we have this unfunded liabilities because the commonwealth for many, many years did not pay its bill. And so we now have to make up for the the lack of work that was done in the in the past. So it doesn't matter what the design is, we have a big liability. And that's going to be there no matter what kind of plan we come up with. PJ from Edgeley says when you were being elected, one of your key points was you would place taxes on the fracking industry here in Pennsylvania, and yet nothing has passed so far. So what are you doing to get any bills passed that will place taxes on these fracking and pipeline industries? Well, I've been working each of my budgets. I proposed a severance tax and in the first two years, they were not accepted by the legislature. I again proposed a severance tax in this budget. And my hope is that that eventually the legislature will agree with the majority of Pennsylvanians that we need a severance tax. In the meantime, I'm also working to to talk to the the industry and say, you know, this is really something you ought to be for. You want people of Pennsylvania to share in the benefits of having this natural resource beneath our feet and and applaud you for for extracting it. And we can do that if if some of the money that that you're making goes to fund our schools. And by the way, Pennsylvania is the only major natural gas producing state without a severance tax. So there are a lot of good reasons why we need to get this. And I'm hoping that maybe this year the legislature will see reason. And talk about what you're proposing this year. This year in my budget proposal, it was a six and a half percent tax on the market value at the well head. But you get 100 percent minus 100 percent credit for the impact fee, which gas drillers are already paying. So the it's a reasonable tax. I understand John Casey, Governor Casey of Ohio, has proposed a much higher severance tax. So we really ought to do this. It would it would benefit our children, our schools and be a nice aid to our budget. Um, let's talk a little bit about animal abuse. That's something if you tuned into the local news at all here in Harrisburg over the last several months, we've heard a lot about House Bill 869. It's formally called Libra's law. And Chris asked, is there any focus on HB 869 if that it makes it to your desk? And I'm sorry, I'm going to paraphrase this one. He says, is there more support across the board this time for that Libra's law? And will you sign it if you get it at your desk? Well, two questions. First, yes, I will sign it if it gets to my desk. And I sort of thought it was going to get to my desk last time around, because I think there should be very broad support for this. This makes perfect sense. So I'm not sure why it didn't get to my desk, but yes, I will sign it if it gets there. OK. All right. Back to your budget and a big one in your budget. Eric from Warren says with the consolidation of the human services departments, such as the Department of Aging, what will happen to the area agencies on aging in all the counties? Will they stay in place as is? Will many people be put out of work? And will the elderly be left without advocates? Absolutely not. The consolidation of the four different agencies is an effort to actually improve services. So the area agencies on on aging, I think, should see and their clients should see better services as a result of this. And I look back at the things we've done best for our citizens have been where agencies have actually worked together like the opioid crisis, substance use disorder crisis. That's actually a partnership of the Department of Human Services, Department of Health, Department of Aging and Department of Drug and Alcohol programs. All those agencies work together. And I think have done a good job in helping Pennsylvania get his arms around the substance use disorder, the opioid epidemic. And I see the same thing for for aging. The Department of Aging is a strong champion for the elderly. We need that kind of championing of the elderly in Pennsylvania. I'm trying to figure out a way to do it even better than we're doing it right now. And I think I think this consolidation will help us do that. And another one we've gotten a lot of questions about regarding this year's budget. Lonnie from Sweet Valley, he's asking about the $25 fee to fund the Pennsylvania State Police that cover municipal areas that don't have their own police forces. That's something that you proposed in this budget. Just talk about that a little bit and the reasoning behind why the Commonwealth needs to go this direction. Sure. There are two things. First of all, this is something that more and more municipalities are opting for and who can blame right now? The rule is that if you have your own police department and you want to move to the state police coverage, you can do that for free. So you reduce your own expenditures by eliminating your police department and simply move to the state police coverage. And we all know there's nothing that's free. And so that that's a problem for everybody else who is helping to pay for defray the cost of that. Second thing is the way that cost is defrayed is that we end up with fewer bridges and roads that are repaired or built. The money that that is being used by the Pennsylvania State Police to fund this free service, this coverage, is basically coming out of act 89 money, money that happened, a tax that was put in place before I came into office. But it means that that every municipality that signs up for this free state police service, the rest of us, all of us get fewer bridges and fewer roads repaired. So it's not sustainable. We need to move on. And $25 per head is a very, very small charge compared to what municipalities with their own police forces are paying. And that's the municipality would pay that? Yeah, but presumably they would they would pass it through as tax to their citizens. But one way or another, all of us are paying for this coverage, whether through the sales tax or the income tax or the opportunity tax that we all the opportunity foregone of having bridges and roads that are up to snuff. Onto the opioid crisis that we've talked a lot about over the last you know, several years here, Jeannie from Neutrona Heights had asked about concerns that, again, with the consolidation of the Department of Drug and Alcohol programs into a larger health and human services agency. Will that make things harder? Will that, you know, kind of stall efforts to fight this epidemic? No, if I knew what you'd say, you know, if I thought if I thought that we're going to were to would make things harder, I wouldn't have done it. The the idea behind breaking down these agency barriers, these these walls and silos is to actually improve service. And I look at what we've done so far in the opioid epidemic. And I think we've done some good. We have a lot of work to do, but but we've gotten some good things done. And that's because people in the Drug and Alcohol Department of Drug and Alcohol programs and the Department of Human Services and Health and Aging, they've all worked together. And and they've they've focused on what's right for the citizen rather than, you know, how does this support some body who decided who knows how many years ago that we ought to have a separate agency for this and a separate agency for that? I want to help the citizens of Pennsylvania. I want to do what's best for the citizens rather than what's best for some accident of history. Continuing on that topic, Julia says, are there any new plans to manage the rising overdose rates in the state? Narcan availability is a start, but does the state intend to address the addiction issue itself with rehab facilities, community outreach? We must do more to solve the problem rather than just treat the symptoms. Do you agree? I agree, absolutely. And I think if you look at what we've been doing, we created 45 Centers of Excellence treatment centers last year. The last ones are just getting up and running now. So this will be the first calendar year where we basically had them up and running. And the hope is that they will be able to address the needs of 11,000 Pennsylvanians who are suffering from substance use disorder. But we need to do more. We need to have more treatment centers. We need to have more options for suffers of substance use disorder. We need to do more to give medical professionals more resources. Naloxone is Narcan is just one thing. It brings a person back from an overdose. But then you have to make sure that that person has treatment options or you haven't done really anything. So we have a lot of work to do. But I put 10 million extra dollars into my budget this year on top of the 20 million additional dollars I put in last year. And with bipartisan support in both the Senate and the House, I believe Pennsylvania is one of the leaders in the country in terms of trying to address this. One other thing, we need to address the stigma of this. This is a disease. And I think one of the things about bringing this out into the open is that we're saying this is a disease. We know how to treat it. There should be no one who dies of this disease in Pennsylvania. Yeah, you're very passionate about this topic. You've made that clear. All right, let's lighten things up just a little. We're live here on Facebook with Governor Wolf answering your questions. And we had one come in from Pennsylvania mayor of Coriopolis. I guess he goes, but Tony Celeste. Yeah, and he says, Governor Wolf, make a stop in Coriopolis. OK. Thanks for the invitation. Well, we'll get it on the schedule. All right, let's move on to Erica from Downing Town. She wants to know where we are in terms of equality for all Pennsylvanians in the area of employment, health care, housing, public accommodations, as it relates to LGBTQ community. And I'm just going to read what she wrote to because I think it's important. Every Northeastern state has anti-discrimination against sexual orientation and many don't allow discrimination against gender identity or expression. P.A. has neither. Are we behind? Yes, we have a ruling federal court that goes back May of 2014 that allows for same-sex marriage and marriage equality in Pennsylvania. But Pennsylvania still does not have legislation on non-discrimination. And I think that is appalling. It is Pennsylvania really has to move in, not just to be with other states, but because it's it's the right thing to do. We need to have non-discrimination legislation. There are some good bills in the General Assembly, and I'm hoping that they will pass as soon as they get to my desk. I will sign them. We're going to hurry along the last couple here because we're running out of time. We have covered a lot of ground. Joseph from Wilkes-Barre says, you mentioned a retirement incentive to reduce the state employee workforce. What is the incentive and will that help pension reform? The what is it has not been worked out and will be worked out with employees and the legislature takes legislative action to make a presentation. But obviously it's one that's going to have to be fair to the employees or they won't want to take it. And that's the nature of early retirement. It's voluntary. It's up to the employee and we'll do our best to make it one that is attractive. It probably won't make much of a difference in terms of the unfunded liability, but it will actually help in terms of the immediate short-term expenses, actually the long-term expenses, in terms of helping us streamline government to figure out how we can get to the point where we're actually doing more for our citizens with less. Early retirement is a portion of that strategy. Adam from Altoona says, Governor Wolff, have you thought about the financial benefits of a statewide veterans court? Yeah, I'm not the way the should be a better constitutional scholar, but I think the way we do it, the nature of courts are really under the direction of the counties. And so the way it's been done so far is that each county determines whether it's going to have a drug court. One of the things in my budget is to provide financial assistance to counties who are considering drug courts. So I think they're a good thing. And I think the people who have gone through drug courts or the people in the legal profession believe that these are good things. But I don't think I could do that on the... I don't think I could have a state drug court, but I'll certainly look into it. It's good to get input from our Facebook fans. Yes. Judah from the Poker's Player's Alliance says, Thank you for the opportunity to ask about the budget's reliance on revenue from online poker. When? When. That's always the good one. When will legalization and regulation occur? Yeah, I don't know. I think, again, there is real work being done in the General Assembly as to what enhanced gambling, including online gambling, including possibly online poker, would look like moving forward. The deal is that whatever we do on this enhanced gaming, it should not take business away from their casinos or the lottery, or otherwise, why would we do it? Because we're already getting revenues from that. It should be something that can be implemented cleanly and that the oversight would be... we could do without being really that intrusive. So there are a lot of details that have to go into this. I know other states have done online poker and have done it successfully. We need to keep learning what we can from those states. I think we are out of time. So we've covered a lot of ground today and unfortunately, again, we are running out of time. I want to thank you so much for joining us here today and thank you for logging on for the Governor's latest Facebook Town Hall. And of course, to learn more about Governor Wolf's budget proposal, don't forget to follow him on Facebook and Twitter. He's very active on there. I appreciate that. Thank you very much for having me on. Thank you.


Hi everyone, we are going to take a look at a couple of examples of how we can use linearization to do some basic arithmetic. Our first one we're going to look at is we want to figure out what 1.97 cubed is, and we can actually use a little bit of calculus to do that. Certainly, you could sit there and do 1.97 times 1.97 times 1.97, try doing that without a calculator. I don't think you'd like to. So let's look at how we can use the linearization process that we've been talking about to find a pretty good estimation of this value. The first thing we need to determine is the function that's going to be the basis for our linearization. Recall what the linearization equation states. It states that l of x is going to be equal to f of a plus f prime of a times the quantity x minus a. And we've talked about that's really simply a variation on the point slope form of the equation of a line. Since our equation is based upon a particular function, we need to determine a function to use for this. Now since we're cubing a value, it might make sense to use x cubed as our function. We know we're going to have to find the derivative of that, so let's go ahead and do that. That simply is 3x squared. We will need to evaluate that at a particular x value. Now we want to do 1.97 cubed. And if you think back to the whole idea behind linearization, it's the idea that we're using a value close to the targeted x. So since we're targeting 1.97 and we want something close to that, that will be easy to cube. Let's make our a value b2. We need then to do f prime of 2 and that gives us 12. Another thing we're going to need is the actual function value at 2. That simply is going to be 8, 2 cubed. So now we have everything to substitute into our linearization equation. Our linearization then states that L of x will be f of 2, so that was 8, plus f prime of 2, that was 12 times the quantity x minus 2. If you distribute and simplify that, you find that we have a linearization equation of 12x minus 16. Now remember the whole idea behind linearization, it's to allow us to use a simple linear equation to obtain a more complicated value. We saw that in earlier videos that you've watched. If we want then to approximate 1.97 cubed, notice I said approximate, we can do that by substituting 1.97 in place of our x in the linearization equation. So we'd want to do 12 times 1.97, take away 16, when you do that you should get 7.64. So that would be a pretty close approximation for the true value of 1.97 cubed. Go ahead and try it on your calculator, do actually 1.97 cubed and you should notice your answer is pretty close to 7.64. Let's take a look at another example. Now we want to determine square root of 24. We definitely can estimate it, we know it's going to be in between 4 and 5 because 4 squared of course is 16, 5 squared is 25. So it's probably going to be closer to 5 than it would be 4 by the time we're done. Once again we're going to use linearization to do that. As we saw in the last example, we need a function that is going to serve as the basis in our linearization as well as the a value that we will use in the linearization as well. Because we're trying to do square root of 24, let's use the function square root of x. And we need an a value to use, we want that a value to be close to 24 and something that would be nice to take the square root of. Let's use 25. We need to find our derivative which simply is going to be 1 half x to the negative 1 half. We can rewrite that as 1 over 2 square root of x. We need to evaluate that derivative at 25. When you do that you get 1 tenth. Of course we need the original function value at 25, that's simply going to be 5. As we substitute into our linearization we get that L of x is equal to f of 25, remember that's simply 5, plus the derivative value, 1 tenth, times the quantity x minus the a value of 25, distribute and simplify and you should obtain 1 tenth x plus 2.5. We can then use this to approximate the true value of square root of 24. You would have 1 tenth times 24 plus 2.5, that ends up to be 2.4 plus 2.5 and we get 4.9. Remember how we had said earlier that our guess is that it would be closer to 5 than it would be 4. If you go ahead and actually do this on your calculator, again you should find that the true value of square root of 24 is pretty close to 4.9.