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 6.75 feet/second
C. about 1.69 feet/second
D. about 4.00 feet/second

To estimate the average rate of change of the elevator's speed function between 3.9 seconds and 8.2 seconds, follow these steps:

1. Understand the Function: We are given the function [tex]$ f(x) = 1.6875x $[/tex], which models the elevator's speed in feet per second, where [tex]$ x $[/tex] is the time in seconds.

2. Determine Time Intervals:
- Initial time ([tex]$ x_1 $[/tex]) is 3.9 seconds.
- Final time ([tex]$ x_2 $[/tex]) is 8.2 seconds.

3. Calculate Function Values:
- Calculate the speed at the initial time:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \text{ feet per second}
\][/tex]
- Calculate the speed at the final time:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet per second}
\][/tex]

4. Find the Average Rate of Change:
- The average rate of change of the function between two points is calculated using the formula:
[tex]\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\][/tex]
- Substitute the known values into the formula:
[tex]\[
\text{Average Rate of Change} = \frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} \approx 1.69 \text{ feet per second}
\][/tex]

5. Final Answer:
- The average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is approximately 1.69 feet per second.

This matches the option: about 1.69 feet/second.