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Answer :
To find the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds, we can follow these steps:
1. Identify the Function: The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] is the time in seconds.
2. Evaluate the Function at the Given Points:
- First, calculate the speed at 3.9 seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \text{ feet per second}
\][/tex]
- Next, calculate the speed at 8.2 seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet per second}
\][/tex]
3. Compute the Average Rate of Change:
The average rate of change of the function over an interval is found by taking the difference in the function values at the endpoints of the interval and dividing by the difference in the endpoints. In this case:
[tex]\[
\text{Average rate of change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
Substituting the known values:
[tex]\[
\text{Average rate of change} = \frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} = 1.6875 \text{ feet per second}
\][/tex]
4. Round the Final Answer:
We want the final answer rounded to two decimal places, so the average rate of change is approximately:
[tex]\[
1.69 \text{ feet per second}
\][/tex]
Therefore, the estimated average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about [tex]\( \mathbf{1.69} \)[/tex] feet per second.
1. Identify the Function: The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] is the time in seconds.
2. Evaluate the Function at the Given Points:
- First, calculate the speed at 3.9 seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \text{ feet per second}
\][/tex]
- Next, calculate the speed at 8.2 seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet per second}
\][/tex]
3. Compute the Average Rate of Change:
The average rate of change of the function over an interval is found by taking the difference in the function values at the endpoints of the interval and dividing by the difference in the endpoints. In this case:
[tex]\[
\text{Average rate of change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
Substituting the known values:
[tex]\[
\text{Average rate of change} = \frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} = 1.6875 \text{ feet per second}
\][/tex]
4. Round the Final Answer:
We want the final answer rounded to two decimal places, so the average rate of change is approximately:
[tex]\[
1.69 \text{ feet per second}
\][/tex]
Therefore, the estimated average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about [tex]\( \mathbf{1.69} \)[/tex] feet per second.
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