roasted turkey cooling

Question

This exercise uses Newton's Law of Cooling.

A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F.

(a) If the temperature of the turkey is 145°F after half an hour, what is its temperature after 45 min? (Round your answer to the nearest whole number.)

(b) After how many hours will the turkey cool to 100°F? (Round your answer to one decimal place.)

 hr

Answer 1

Let

E = 75°F be the temperature of the environment,

T(t) be the temperature of the turker after t minutes.

We are given

T(0) = 185°F

The Newton's Law of Cooling states

T(t) = E +(T(0) - E)e^-rt (1)

for some unknown constant r, which depends on the turkey.

(a)

We are given

T(30) = 145°F (2)

We are to calculate T(45).

First calculate r from (1) and (2):

T(30) = E + (T(0) - E)e^-30r

ln(T(30) - E) = ln(T(0) - E) - 30r

r = (ln(T(30) - E) - ln(T(0) - E))/(-30)

= (ln(145 - 75) - ln(185 - 75))/(-30) = 0.015

Now we can calculate T(45) from (1)

T(45) = E +(T(0) - E)e^-45r

= 75 + (185 - 75)e^-45x0.015 = 130°F

(b)

We are to find time t such that T(t) = 100°F

Substitute all the known quantities into (1)

100 = 75 + (185 - 75)e^-0.015t

e^-0.015t = 25/110

-0.015t = ln(25/110)

t = ln(25/110)/(-0.015) = 99 min