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------------------------------------------------ The speed of an elevator (in feet per second) is modeled by the function [tex]f(x) = 1.6875x[/tex], where [tex]x[/tex] is time in seconds. Estimate the average rate of change between 3.9 seconds and 8.2 seconds. Round the final answer to two decimal places.

A. about 6.75 feet/second
B. about 0.59 feet/second
C. about 4.00 feet/second
D. about 1.69 feet/second

Answer :

To estimate the average rate of change of the elevator's speed function [tex]\( f(x) = 1.6875x \)[/tex] between 3.9 seconds and 8.2 seconds, follow the steps below:

1. Find the speed at 3.9 seconds:
Use the function [tex]\( f(x) = 1.6875x \)[/tex] to calculate the speed when [tex]\( x = 3.9 \)[/tex]:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58 \, \text{feet/second (approximately)}
\][/tex]

2. Find the speed at 8.2 seconds:
Again, use the function to find the speed when [tex]\( x = 8.2 \)[/tex]:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.84 \, \text{feet/second (approximately)}
\][/tex]

3. Calculate the average rate of change:
The average rate of change of the function from 3.9 seconds to 8.2 seconds is given by the formula:
[tex]\[
\frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
Substitute the values found:
[tex]\[
\frac{13.84 - 6.58}{8.2 - 3.9} = \frac{7.26}{4.3} = 1.69 \, \text{feet/second (rounded to two decimal places)}
\][/tex]

Therefore, the estimated average rate of change between 3.9 seconds and 8.2 seconds is about 1.69 feet/second.