Chapter 2.2, Problem 11E

Baking a Potato: A potato is placed in a preheated oven to bake. Its temperature $P=P\left(t\right)$ is given by

$P=400-325e^{-t/50}$,

Where P is measured in degrees Fahrenheit and t is the time in minutes since the potato was placed in the oven.

a. Make a graph of P versus t.(Suggestion: in choosing your graphing window, it is reasonable to look at the potato over no more than a 2-hour period. After that, it will surely be burned to a crisp. You may wish to look at a table of values to select a vertical span.)

b. What was the initial temperature of the potato?

c. Did the potato’s temperature rise more during the first 30 minutes or second 30 minutes of baking? What was the average rate of change per minute during the first 30 minutes? What was the average rate of change per minute during the second 30 minutes.

d. Is this graph concave up or concave down? Explain what that tells you about how the potato heats up, and relate this to part c..

e. The potato will be done when it reaches a temperature of 270 degrees. Approximate the time when the potato will be done.

f. What is the temperature of the oven? Explain how you got your answer. (Hint: if the potato were left in the oven for a long time, its temperature would match that of the oven.)