Question

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?

Answer 1

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

Answer 2

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

Answer 3

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 \(\frac{dT}{dt}=\frac{ \Delta T}{\Delta t}\)

Answer 4

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. :)

Answer 5

found this http://learn.flvs.net/webdav/educator_apcalcab_v14/sfframe.htm?loc=http%3A%2F%2Flearn.flvs.net%2Feducator%2Fstudent%2Fframe_toolbar.cgi%3Fdhicks59*jdosio1*mpos%3D1%26spos%3D0%26option%3Dhidemenu%26slt%3DLjph6mFdLTT1E*4063*http%3A%2F%2Flearn.flvs.net%2Fwebdav%2Feducator_apcalcab_v14%2Findex.htm&ltiloc=http%3A%2F%2Ftool.studyforge.net%2Flti.php%3Fresource_type%3Dlesson%26resource_id%3D4715

Answer 6

that link is no good

Answer 7

do you know how to solve the integrals?

Answer 8

yes i do

Answer 9

OK, so whats the next step?

Answer 10

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

Answer 11

i mean right

Answer 12

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

Answer 13

\[T = 105e ^{-0.14t} + 75\]

Answer 14

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

Answer 15

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

Answer 16

yep. thats it. now solve for t