I encountered this Newton's Law of Cooling questions in my textbook and usually it gives the temperature when t = 0, but this time it didn't, which is giving me a hard time.
It says exactly as follows,
A hot object is placed in a room whose temperature is 80F. After 1 minute, the temperature of the object is 160F and its rate of change of temperature is 22F per minute. Find the initial temperature of the object.
Here is my attempt,
dT/dt = k (t - 80)
dT(1/(T-80) = dt k
ln (T-80) = kt +c
T = ce^(kt) + 80
T prime = [ce^(tk)]/k
So, it says
T(1) = 160 F
T prime(1) = 22 F/min
Which gives me,
160 = ce^k + 80
22 = (ce^k)/k
I solved for c in the second eq.
22k/e^k = c
Plug in the first eq.
160 = (22k/e^k) * e^k + 80
80 = 22k
k = 80/22
So I have k now, now I solve for c,
160 = ce^(80/22) + 80
c = 80/e^(80/22)
This all seems correct to me but, BUT when I test it out on my calculator, the values it gives seems way too large.
The textbook has no solution for even numbers, so I am unable to compare answers.
Can you guys spot my mistake...?
Check your T prime!
(Sorry, this would work better as a comment, but I need one upvote for that)