## anonymous one year ago Really need some help understanding this problem?!?! The temperature of a cup of coffee varies according to Newton's Law of Cooling: dT/dt = -k (T-A), where T is the water temperature, A is the room temperature, and k is a positive constant. If the coffee cools from 180°F to 100°F in 10 minutes at a room temperature of 75°F, how long (to the nearest minute) will it take the water to cool to 80°F?

1. Empty

Ahhh this is a fun one. Any ideas, how far can you get on your own just doing anything? What have you tried?

2. anonymous

Well in know that it has something to do with the separation of variables, which is just a simple way of solving for a basic differential equation. But the issue lies in interpreting the numbers given and replacing them into the equation to make it solvable. in short, I have nothing

3. Empty

Well I think we can do this with an integrating factor or possibly some other way too, this is just an ODE not a PDE so it should be easier to solve. Since we're rounding it's even possible to just estimate some things such as replacing $$\frac{dT}{dt}=\frac{ \Delta T}{\Delta t}$$

4. Empty

Ok so here's what we do, substitute in: $u=T-A$ differentiate to get: $\frac{du}{dt} = \frac{dT}{dt}$ this just simplifies the differential equation to become: $\frac{du}{dt} = -ku$ which is separable: $\int \frac{du}{u}=\int -kdt$ Now you can solve this for T(t), so do this now and I'll help you out. :)

5. anonymous
6. alekos

that link is no good

7. alekos

do you know how to solve the integrals?

8. anonymous

yes i do

9. alekos

OK, so whats the next step?

10. anonymous

this is what i'v got so far the left side is likely wrong

11. anonymous

i mean right

12. alekos

Thats all really good! You've found k = 0.14 and C = 105 so now you have the full equation and you can use this to find t when the temperature drops to 75 deg

13. alekos

$T = 105e ^{-0.14t} + 75$

14. alekos

I meant find t when the temp drops to 80. So substitute T=80

15. anonymous

did that, was wrong. by the way T = 80 as final temp so it would be $80 = 105e^(-.14t)+75$

16. alekos

yep. thats it. now solve for t