Answer :
To estimate the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds, we can use the following steps:
1. Understand the Function: The function given is [tex]\( f(x) = 1.6875x \)[/tex]. This function represents the speed of the elevator in feet per second, where [tex]\( x \)[/tex] is the time in seconds.
2. Identify the Interval: We're looking at the time interval from 3.9 seconds to 8.2 seconds.
3. Find the Function Values:
- Calculate the speed at 3.9 seconds: [tex]\( f(3.9) = 1.6875 \times 3.9 \)[/tex].
- Calculate the speed at 8.2 seconds: [tex]\( f(8.2) = 1.6875 \times 8.2 \)[/tex].
4. Calculate the Change in Speed:
- The change in speed (change in function values) is [tex]\( f(8.2) - f(3.9) \)[/tex].
5. Calculate the Change in Time:
- The change in time is [tex]\( 8.2 - 3.9 \)[/tex].
6. Find the Average Rate of Change:
- The average rate of change is the change in speed divided by the change in time:
[tex]\[
\text{Average rate of change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
7. Round the Result:
- Round the result to two decimal places to ensure precision.
After performing these steps, we find that 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.
1. Understand the Function: The function given is [tex]\( f(x) = 1.6875x \)[/tex]. This function represents the speed of the elevator in feet per second, where [tex]\( x \)[/tex] is the time in seconds.
2. Identify the Interval: We're looking at the time interval from 3.9 seconds to 8.2 seconds.
3. Find the Function Values:
- Calculate the speed at 3.9 seconds: [tex]\( f(3.9) = 1.6875 \times 3.9 \)[/tex].
- Calculate the speed at 8.2 seconds: [tex]\( f(8.2) = 1.6875 \times 8.2 \)[/tex].
4. Calculate the Change in Speed:
- The change in speed (change in function values) is [tex]\( f(8.2) - f(3.9) \)[/tex].
5. Calculate the Change in Time:
- The change in time is [tex]\( 8.2 - 3.9 \)[/tex].
6. Find the Average Rate of Change:
- The average rate of change is the change in speed divided by the change in time:
[tex]\[
\text{Average rate of change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
7. Round the Result:
- Round the result to two decimal places to ensure precision.
After performing these steps, we find that 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.
