Maths problem: Where to put the alarm clock?

2011 May 11
by Joonas Mäkinen

The purpose of applied mathematics is to create models. Mathematical models form a framework for understanding how any mechanical, social, biological or whatever system works. Obviously this is serious business because the better we can predict how different variables go together, the better we can prepare outselves for trouble and maybe even control the phenomenon in question to make it do our bidding.

Figuring out how fast glaciers are melting, when the next earthquake is threatening us, why skin wrinkles after taking a bath or how we could possibly kill that darn virus that causes epidemic fevers is important: we can save lives and natural resources, create better services and products. But mathematics is not about just large industry applications – it’s about everything.

As an anti-climactic example of how everyday events could be studied and improved through modelling, here’s a non-rigorous, overkill optimisation problem I thought about some time ago, presented in a very boring mathsy exercise way:

Assuming the alarm clock is placed distance S away from the bed, the intensity I of the ringing alarm is inversely proportional to the square of the distance, the minimum perceived noise signal needed to wake up is I₀, the perceived signal being proportional to the logarithm of the stimulant I, level of awakeness being proportional to time awake, there existing a set level of awakeness A₀, after which one doesn’t fall asleep again after waking up and closing the alarm, and the person walking the distance 2S with constant velocity V, how far from the bed should the alarm clock be placed to maximise the probability of waking up and staying awake?

That was one sentence. One sentence that combines several models known from psychology and acoustics. I made up the part about awakeness – that was just me trying to model the situation, thinking that a directly proportional model could approximate it well. The problem here of course is that if the alarm is right next to you, it’s louder and wakes you up better but it can be also turned of quickly, lulling you back to sleep. Needing to walk over to it wakes you up but from far away the alarm also rings more faintly. Somewhere between there lies an optimal distance. (This doesn’t yet take into account that a very sudden and loud alarm does wake you up “better”. Maybe I’ll work this into a bigger study, who knows…)

Do tell if you manage to improve the given assumptions and figure out the solution. Or a solution. If it is even solvable.