Chapter 4.2, Problem 59PE

Newton’s law of cooling indicates that the temperature of a warm object, such as a cake coming out of the oven, will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T\left(t\right)$

is modeled by $T\left(t\right)=T_0+\left(T_0-T_a\right)e^{-kt}$

. In this model, $T_a$

represents the temperature of the surrounding air, $T_0$

represents the initial temperature of the object, and t is the time after the object starts cooling. The value of k is a constant of proportion relating the temperature of the object to its rate of temperature change. Use this model for Exercises 59–60.

A cake comes out of the oven at $350°\text{F}$ and is placed on a cooling rack in a $78°\text{F}$ kitchen. After checking the temperature several minutes later, the value of k is measured as 0.046.

a. Write a function that models the temperature $T\left(t\right)\left(\text{in }°\text{F}\right)$ of the cake t minutes after being removed from the oven.

b. What is the temperature of the cake 10 min after coming out of the oven? Round to the nearest degree.

c. It is recommended that the cake should not be frosted until it has cooled to under $100°\text{F}$. If Jessica waits 1 hr to frost the cake, will the cake be cool enough to frost?