# 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.

### Precalculus: Mathematics for Calcu...

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

### 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

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=T2−T1 Rate of temprature decrease=T2−T1t2−t1 t(min) Decrease in temperature=T2−T1 Rate of temprature decrease=T2−T1t2−t1 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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