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------------------------------------------------ Average Rate of Change Quick Check

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 0.59 feet/second
B. about 400 feet/second
C. about 1.69 feet/second
D. about 6.75 feet/second

Answer :

To find the average rate of change of the function [tex]\( f(x) = 1.6875x \)[/tex] between 39 seconds and 8.2 seconds, we follow these steps:

1. Identify the time values:
We have [tex]\( x_1 = 8.2 \)[/tex] seconds and [tex]\( x_2 = 39 \)[/tex] seconds.

2. Calculate the function values at these points:
- For [tex]\( x_1 = 8.2 \)[/tex]:
[tex]\[
f(x_1) = 1.6875 \times 8.2
\][/tex]
This equals approximately 13.84 feet.

- For [tex]\( x_2 = 39 \)[/tex]:
[tex]\[
f(x_2) = 1.6875 \times 39
\][/tex]
This equals approximately 65.81 feet.

3. Calculate the average rate of change:
The average rate of change is found by taking the difference in the function values and dividing by the difference in time:
[tex]\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
Plugging in the values we calculated:
[tex]\[
\frac{65.81 - 13.84}{39 - 8.2} = \frac{51.97}{30.8} \approx 1.69 \text{ feet per second}
\][/tex]

Therefore, the average rate of change between 8.2 seconds and 39 seconds is about 1.69 feet per second.