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.
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.
