Concept explainers
Cooling Soup When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature T of the soup is a function of time t. The table below gives the temperature (in °F) of a bowl of soup t minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly?
t (min) | T (°F) |
0 | 200 |
5 | 172 |
10 | 150 |
15 | 133 |
20 | 119 |
25 | 108 |
30 | 100 |
35 | 94 |
40 | 89 |
50 | 81 |
60 | 77 |
90 | 72 |
120 | 70 |
150 | 70 |
To find: The average rate of change of temperature of soup over the first 20 minutes and over the next 20 minutes also during which interval the soup cool off more quickly.
Answer to Problem 29E
Therefore, the average rate of change of temperature of bowl of soup over the first 20 minutes is
Explanation of Solution
Given:
The given table is,
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0 | 200 | 35 | 94 | |
5 | 172 | 40 | 89 | |
10 | 150 | 50 | 81 | |
15 | 133 | 60 | 77 | |
20 | 119 | 920 | 72 | |
25 | 108 | 120 | 70 | |
30 | 100 | 150 | 70 |
Formula used:
Average rate of change of function
Calculation:
The table shows the temperature (in
Temperature of bowl at time 0 min is 200
Substitute 200 for
The average rate of change of temperature of bowl of soup over the first 20 minutes is
Substitute 119 for
The average rate of change of temperature of bowl of soup over the next 20 minutes is
Where,
Form a table to show the decrease in temperature during each interval of time in the given table.
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0-5 | 28 | 5.6 | 35-40 | 5 | 0.5 | |
5-10 | 22 | 4.4 | 40-50 | 8 | 0.8 | |
10-15 | 17 | 3.4 | 50-60 | 6 | 0.6 | |
15-20 | 14 | 2.8 | 60-90 | 5 | 0.166 | |
20-25 | 11 | 2.2 | 90-120 | 2 | 0.066 | |
25-30 | 8 | 1.6 | 120-150 | 0 | 0 | |
30-35 | 6 | 1.2 |
From the table it is clear that during first interval the soup cool off more quickly.
Therefore, the average rate of change of temperature of bowl of soup over the first 20 minutes is
Chapter 2 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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