Expert: Paul Klarreich - 5/30/2007

Question
Newton's law of cooling states that the rate at which an object cools is directly proportional to the difference in temperature between the objects and its surroundings.
- It is 9:00 PM. The room temperature is 68 degrees when you pour a glass of 40 degree milk. by 9:03 the milk has warmed to 43 degrees and the phone rings. Distracted by the conversation, you forget about the glass of milk. If you dislike milk warmer than 60 degrees how long, to the nearest minute, do you have to solve the calculus problem and still enjoy acceptably cold milk with your cookies.

Answer
Questioner:   Erica
Category:  Calculus
Question:  Newton's law of cooling states that the rate at which an object cools is directly proportional to the difference in temperature between the objects and its surroundings.
- It is 9:00 PM. The room temperature is 68 degrees when you pour a glass of 40 degree milk. by 9:03 the milk has warmed to 43 degrees and the phone rings. Distracted by the conversation, you forget about the glass of milk.

If you dislike milk warmer than 60 degrees how long, to the nearest minute, do you have to solve the calculus problem and still enjoy acceptably cold milk with your cookies.
............................
Hi, Eri,

Sorry, I mean

Hi, Erica,

[I thought you were someone else.]

Let  T = temp difference in minutes.
Let  t = time starting at 9 PM.

Here's your law of cooling:

dT/dt = kT

dT
-- = k dt
T

ln T = kt + c

T = e^(kt + c)

T = C e^kt    <<  Big C is not the same as little c.

Now put in a bit of data:

At t = 0, T = 28 degrees.

T(0) = C e^k(0)
28 = C

Ok, now:

T = 28 e^kt

Also:  at  t = 3, T = 25

25 = 28 e^k(3)

25 = 28 e^3k

3k = ln(25/28)
k = ln(25/28)/3

Use your calculator to find:

k = -0.037776228435667724912766106635719

Call it  -0.0378

So your equation is:

T = 28 e^-0.0378t

Only thing left:  Set  T = 8 (degrees) and solve for t.

8 = 28 e^-0.0378t

-0.0378t = ln(8/28)

t = - ln(8/28)/ 0.0378
t =   ln(28/8)/ 0.0378

Use the calculator for that.  I get:

33.141877473422433748362979417593  minutes.

or  33 minutes, to nearest...