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Answer :
To estimate the average rate of change of the speed of an elevator between 3.9 seconds and 8.2 seconds, we follow these steps:
1. Identify the function for speed: The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] represents the time in seconds.
2. Calculate the speed at the starting time:
- Plug [tex]\( x = 3.9 \)[/tex] into the function to find the speed at 3.9 seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \text{ feet/second}
\][/tex]
3. Calculate the speed at the ending time:
- Plug [tex]\( x = 8.2 \)[/tex] into the function to find the speed at 8.2 seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet/second}
\][/tex]
4. Determine the average rate of change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
- Substitute the respective values:
[tex]\[
\frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} \approx 1.6875 \text{ feet/second}
\][/tex]
5. Round the result:
- The average rate of change, rounded to two decimal places, is approximately 1.69 feet/second.
Thus, the average rate of change in the speed of the elevator between 3.9 seconds and 8.2 seconds is about 1.69 feet per second.
1. Identify the function for speed: The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] represents the time in seconds.
2. Calculate the speed at the starting time:
- Plug [tex]\( x = 3.9 \)[/tex] into the function to find the speed at 3.9 seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \text{ feet/second}
\][/tex]
3. Calculate the speed at the ending time:
- Plug [tex]\( x = 8.2 \)[/tex] into the function to find the speed at 8.2 seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet/second}
\][/tex]
4. Determine the average rate of change:
- The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
- Substitute the respective values:
[tex]\[
\frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} \approx 1.6875 \text{ feet/second}
\][/tex]
5. Round the result:
- The average rate of change, rounded to two decimal places, is approximately 1.69 feet/second.
Thus, the average rate of change in the speed of the elevator between 3.9 seconds and 8.2 seconds is about 1.69 feet per second.
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