# newtons cooling law. I need help with this problem. Thank you.

• Oct 13th 2012, 04:53 PM
ALAIN971
newtons cooling law. I need help with this problem. Thank you.
A thermometer reading 70°F is placed in an oven preheated to
a constant temperature. Through a glass window in the oven
door, an observer records that the thermometer read 110°F after
.5 minute and 145°F after 1 minute. How hot is the oven?

using the following:
dT/dt = k(T- Tm)
• Oct 13th 2012, 05:07 PM
chiro
Re: newtons cooling law. I need help with this problem. Thank you.
Hey ALAIN971.

This is a separable DE which means you move the T's to one side and the t's (and the differentials) to the other. Can you separate these variables and their differentials and integrate both sides?
• Oct 13th 2012, 05:30 PM
ALAIN971
Re: newtons cooling law. I need help with this problem. Thank you.
I have done this up to this step.

T(t)=Tm + Ce^(kt)

i get 2 equations with 2 variables but i am stuck.

145= Tm + Ce^(kt)

110= Tm + Ce^(0.5kt)

C and k are constants.
• Oct 13th 2012, 05:31 PM
ALAIN971
Re: newtons cooling law. I need help with this problem. Thank you.
sorry i meant :

I have done this up to this step.

T(t)=Tm + Ce^(kt)

i get 2 equations with 2 variables but i am stuck.

145= Tm + Ce^(k)

110= Tm + Ce^(0.5k)

C and k are constants.
• Oct 13th 2012, 06:12 PM
chiro
Re: newtons cooling law. I need help with this problem. Thank you.
What about the information at t = 0? Can you use this information to get another piece of information?
• Oct 13th 2012, 06:22 PM
ALAIN971
Re: newtons cooling law. I need help with this problem. Thank you.
at t=o temperature is 70
• Oct 13th 2012, 06:31 PM
chiro
Re: newtons cooling law. I need help with this problem. Thank you.
So now you have three equations for Tm, C and k.

70 = Tm + C.
145= Tm + Ce^(k)
110= Tm + Ce^(0.5k)

C = 70 - Tm
k = ln(145 - Tm) - ln(C)
k = 2*ln(110 - Tm) - 2*ln(C)

ln(145 - Tm) = 2*ln(110 - Tm) - ln(C)

145 - Tm = 70 + C
110 - Tm = 40 + C which means

ln(70 + C) = 2*ln(40 + C) - ln(C) which means

ln([70 + C]C) = ln([40 + C]^2) taking exponentials we get:

C[70+C] = [40+C]^2. and how you have a quadratic equation for C.
• Oct 13th 2012, 06:43 PM
ALAIN971
Re: newtons cooling law. I need help with this problem. Thank you.
ok how do i proceed then.
i found C=-160
pit it back in 145-Tm=70+C
and i found Tm=235 Farenheights