BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 2.4, Problem 29E
To determine

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.

Expert Solution

Answer to Problem 29E

Therefore, the average rate of change of temperature of bowl of soup over the first 20 minutes is 4.05°F/min and over next 20 minutes is 1.5°F/min and during first interval the soup cools off more quickly.

Explanation of Solution

Given:

The given table is,

t(min) T(°F)   t(min) T(°F)
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 y=f(x) between x=a and x=b .

Averagerateofchange=ChangeinyChangeinx=f(b)f(a)ba (1)

Calculation:

The table shows the temperature (in °F ) of a bowl of soup t min after it was set on the table.

Temperature of bowl at time 0 min is 200 °F and at time after 20 min 119 °F .

Substitute 200 for f(a) , 119 for f(b) , 0 for a and 20 for b in equation (1).

Averagerateofchange=119200200=8120=4.05°F/min

The average rate of change of temperature of bowl of soup over the first 20 minutes is 4.05°F/min .

Substitute 119 for f(a) , 89 for f(b) , 20 for a and 40 for b in equation (1).

Averagerateofchange=891194020=3020=1.5°F/min

The average rate of change of temperature of bowl of soup over the next 20 minutes is 1.5°F/min .

Rate oftempraturedecrease=T2T1t2t1

Where, T1 is the initial temperature at time t1 and T2 is the successive temperature at time t2 .

Form a table to show the decrease in temperature during each interval of time in the given table.

t(min) Decrease in temperature=T2T1 Rate oftempraturedecrease=T2T1t2t1   t(min) Decrease in temperature=T2T1 Rate oftempraturedecrease=T2T1t2t1
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 4.05°F/min and over next 20 minutes is 1.5°F/min and during first interval the soup cools off more quickly.

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