   Chapter 3.8, Problem 15E

Chapter
Section
Textbook Problem

A roast 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 150°F after half an hour, what is the temperature after 45 minutes?(b) When will the turkey have cooled to 100°F?

(a)

To determine

To find: The temperature after 45 minutes.

Explanation

Given:

The room temperature is Ts=75°F and T(0)=185°F.

Newton’s law of cooling:

The formulate Newton’s law of cooling as a differential equation dTdt=k(TTs), where k is a constant. The change of variable y(t)=T(t)Ts.

Theorem used:

“The only solutions of the differential equation dydx=ky are the exponential functions

y(x)=y(0)ekx.”

Calculation:

Let T be the temperature after the time t.

Substitute Ts=75°F in dTdt=k(TTs),

dTdt=k(T75)

Let y(t)=T(t)75 and substitute t=0 in y(t)

y(0)=T(0)75

Substitute T(0)=185 in the above equation,

y(0)=18575y(0)=110

Since y is a solution of the initial problem with initial condition y(0)=110.

It follows that dydx=ky

Then, the exponential functions y(x)=y(0)ekx.

y(x)=110ekx (1)

The temperature of the turkey is 150°F after half an hour

(b)

To determine

To find: The time that is turkey cooled to 100°F.

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