Newton's Law of Cooling/Warming

[hank]

A thermometer is removed from a room to the outside where the temp is 5 degrees F. After 1 min, the thermometer reads 55 F, and after 5 mins it reads 30 F. What is the initial temp of the room?

Ok, so here's what I know..

dT/dt = k(T - M), and M = 5 F.
T(1) = 55, T(5) = 30

Here's what I want to know...
T(0) = ?

I decided to start off this problem by plugging in numbers and solving for T:

dT/dt = k(T - 5)
dT/|T - 5 | = k dt //by separation of variables
ln |T - 5| = kt + C //after integration
T - 5 = Ce^kt //after exponentiation

Here's where I get stuck. I'm not sure how to find C or k using the information I've been given. The closest I can come up with is..

T(0) = 5 + C
T - 5 = C
T = 5 + (T - 5)e^(kt - 1)

But this doesn't seem right to me.

Thanks in advance,

Hank

 

[Deleted member 4993]

A thermometer is removed from a room to the outside where the temp is 5 degrees F. After 1 min, the thermometer reads 55 F, and after 5 mins it reads 30 F. What is the initial temp of the room?

Ok, so here's what I know..

dT/dt = k(T - M), and M = 5 F.
T(1) = 55, T(5) = 30

Here's what I want to know...
T(0) = ?

I decided to start off this problem by plugging in numbers and solving for T:

dT/dt = k(T - 5)
dT/|T - 5 | = k dt //by separation of variables
ln |T - 5| = kt + C //after integration
T - 5 = Ce^kt //after exponentiation

T(1) = 55

55-5 =Ce^k...............................(1)

T(5) = 30

30-5 = Ce^(5k)..........................(2)

divide (2) by (1)

2 = e^(4k)

k = 1/4 * ln 2............................(3)

Now continue.........

Here's where I get stuck. I'm not sure how to find C or k using the information I've been given. The closest I can come up with is..

T(0) = 5 + C
T - 5 = C
T = 5 + (T - 5)e^(kt - 1)

But this doesn't seem right to me.

Thanks in advance,

Hank

 

[hank]

Thank you very much.

It never occurred to me to divide on equation by the other.
I was able to finish the problem!