### Home > INT3 > Chapter 8 > Lesson 8.2.2 > Problem8-83

8-83.

Guests are arriving for a show when the theater manager realizes he forgot to turn on the air conditioning. The guests are very hot and uncomfortable. At 2:04 p.m. the manager turns on the air conditioning. It is $85^\circ$F in the theater.

1. At 2:15 p.m. the manager announces that he has bad news. Using average rate of change he predicts the theater will not cool off to a comfortable $70^\circ$F until 3:00 p.m. What calculations did the theater manager make? What was the temperature at 2:15 p.m.?

 Time of day (p.m.) Temperature ($^\circ{\text{F}}$) 2:04 2:15 3:00 $85$ $70$
2. As the air conditioner continued cooling, the manager recorded these actual temperatures:

 Time of day (p.m.) Temperature ($^\circ$F) 2:04 2:15 2:24 2:32 2:37 2:53 3:01 3:18 3:27 3:40 $85$ $79$ $78$ $76$ $73$ $72$ $71$ $70$ $70$

What was the average rate of change after 3:00 p.m. until the temperature reached $70^\circ$F? Why was the manager so far off in his prediction?

By looking at the table, the temperature dropped $2^\circ$ in $26$ minutes.