Chapter 5.4, Problem 1TU

TEST YOUR UNDERSTANDING | FOR EXAMPLE $5.10$

When the cake in the example reaches a temperature of $300$ degrees, it is baked. The cake is taken from the oven and placed on a counter to cool. The temperature of the room is $75\text{degrees}$. The difference between the cake’s temperature and room temperature is an exponential function of time, in minutes, with decay factor $0.85$. What is the initial value of this exponential function? What is the limiting value of the cake's temperature? Find a formula that gives the cake's temperature after $t$ minutes.

EXAMPLE $5.10$BAKING A CAKE

A pan of cake batter is initially at a room temperature of $75\text{degrees}$. The pan is placed in a $350-\text{degree}$ oven to bake. Let $C=C\left(t\right)$ denote the temperature of the cake batter $t\text{minutes}$ after it is placed in the oven. In this example, all temperatures are measured in degrees Fahrenheit.

Part 1 The temperature of the cake batter is given by

$C=\text{ Limiting value — Temperature}$ difference.

What is the limiting value of $C$.

Part 2 Let $D$ denote the difference between the oven temperature and that of the batter. Then Newton's law of cooling tells us that $D$ is an exponential function. What is the initial value of $D$.

Part 3 After $10$ minutes, we find that the cake batter has heated to $165$ degrees. Find a formula for $D$.

Part 4 Find a model for the temperature of the cake batter $t$ minutes after it is placed in the oven.