A cold beer initially at 35ºF warms up to 40ºF in 3 min while sitting in a room of temperature 70ºF. How warm will the beer be if left out for 20 min

Question

A cold beer initially at 35ºF warms up to 40ºF in 3 min while sitting in a room of temperature 70ºF.  How warm will the beer be if left out for 20 min

anonymous · Answer

Now if I remember correctly, temperate changes at an exponential rate, because it is a differential equation.

abb0t · Answer

Yep. Go on...

anonymous · Answer

That is $$
\frac{dT}{dt} = kT
$$Where big $T$ is temperature and $t$ is time.  So we need to solve: $$
T(t) = T_0e^{kt}
$$

anonymous · Answer

We know $T_0 = 35^\circ F$ and $T(3) = 40^\circ F$.  We'll let $t$ be in minutes.

anonymous · Answer

So we find $k$ given that:$$ \Large
40 = 35 e^{k(3)}
$$

anonymous · Answer

I think you know how to do the rest.

abb0t · Answer

You're good :)

anonymous · Answer

I'm not that good, it's just I've seen problems like this around here and by now they're relatively easy.

anonymous · Answer

I really need to brush up on my vector calculus though.  Not enough questions like that on this site.

abb0t · Answer

up to multiple integrals or r u referring to stokes theorem, line integrals, and surface integrals?

anonymous · Answer

I recently went over limits, derivatives, and integrals very very closely to get rid of any misconception I might have had.  I wanna do the same with vector functions, multiple integrals, surface integrals, ect.

abb0t · Answer

I can post one or 2 up if you want to try it? I'll help guide you the best I can if you have time right now.

anonymous · Answer

Sure, that would be fun.