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Answer :
We want to find the average rate of change of the function
[tex]$$
f(x) = 1.6875 \, x
$$[/tex]
between [tex]$x = 3.9$[/tex] seconds and [tex]$x = 8.2$[/tex] seconds. The average rate of change is calculated by the formula
[tex]$$
\text{Average Rate} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}.
$$[/tex]
Step 1. Compute [tex]$f(3.9)$[/tex] and [tex]$f(8.2)$[/tex]
- For [tex]$x = 3.9$[/tex] seconds:
[tex]$$
f(3.9) = 1.6875 \times 3.9 = 6.58125.
$$[/tex]
- For [tex]$x = 8.2$[/tex] seconds:
[tex]$$
f(8.2) = 1.6875 \times 8.2 = 13.8375.
$$[/tex]
Step 2. Compute the change in [tex]$f(x)$[/tex]
Subtract the value at [tex]$3.9$[/tex] seconds from the value at [tex]$8.2$[/tex] seconds:
[tex]$$
\Delta f = f(8.2) - f(3.9) = 13.8375 - 6.58125 = 7.25625.
$$[/tex]
Step 3. Compute the change in time
Subtract the smaller time from the larger time:
[tex]$$
\Delta x = 8.2 - 3.9 = 4.3.
$$[/tex]
Step 4. Compute the average rate of change
Divide the change in the function value by the change in time:
[tex]$$
\text{Average Rate} = \frac{7.25625}{4.3} \approx 1.6875.
$$[/tex]
Rounding to two decimal places, the average rate of change is approximately:
[tex]$$
1.69 \text{ feet per second}.
$$[/tex]
Thus, the average rate of change of the elevator’s speed between [tex]$3.9$[/tex] seconds and [tex]$8.2$[/tex] seconds is about [tex]$1.69$[/tex] feet per second.
[tex]$$
f(x) = 1.6875 \, x
$$[/tex]
between [tex]$x = 3.9$[/tex] seconds and [tex]$x = 8.2$[/tex] seconds. The average rate of change is calculated by the formula
[tex]$$
\text{Average Rate} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9}.
$$[/tex]
Step 1. Compute [tex]$f(3.9)$[/tex] and [tex]$f(8.2)$[/tex]
- For [tex]$x = 3.9$[/tex] seconds:
[tex]$$
f(3.9) = 1.6875 \times 3.9 = 6.58125.
$$[/tex]
- For [tex]$x = 8.2$[/tex] seconds:
[tex]$$
f(8.2) = 1.6875 \times 8.2 = 13.8375.
$$[/tex]
Step 2. Compute the change in [tex]$f(x)$[/tex]
Subtract the value at [tex]$3.9$[/tex] seconds from the value at [tex]$8.2$[/tex] seconds:
[tex]$$
\Delta f = f(8.2) - f(3.9) = 13.8375 - 6.58125 = 7.25625.
$$[/tex]
Step 3. Compute the change in time
Subtract the smaller time from the larger time:
[tex]$$
\Delta x = 8.2 - 3.9 = 4.3.
$$[/tex]
Step 4. Compute the average rate of change
Divide the change in the function value by the change in time:
[tex]$$
\text{Average Rate} = \frac{7.25625}{4.3} \approx 1.6875.
$$[/tex]
Rounding to two decimal places, the average rate of change is approximately:
[tex]$$
1.69 \text{ feet per second}.
$$[/tex]
Thus, the average rate of change of the elevator’s speed between [tex]$3.9$[/tex] seconds and [tex]$8.2$[/tex] seconds is about [tex]$1.69$[/tex] feet per second.
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