Answer :
We first calculate the elevator speed at each time using the function
[tex]$$
f(x)=1.6875\,x.
$$[/tex]
At [tex]$x=3.9$[/tex] seconds, the speed is
[tex]$$
f(3.9)=1.6875 \cdot 3.9=6.58125\ \text{feet per second}.
$$[/tex]
At [tex]$x=8.2$[/tex] seconds, the speed is
[tex]$$
f(8.2)=1.6875 \cdot 8.2=13.8375\ \text{feet per second}.
$$[/tex]
The average rate of change over the time interval is given by the formula
[tex]$$
\text{Average rate of change}=\frac{f(8.2)-f(3.9)}{8.2-3.9}.
$$[/tex]
Substitute the computed speeds:
[tex]$$
\text{Average rate of change}=\frac{13.8375-6.58125}{8.2-3.9}.
$$[/tex]
Calculate the differences:
[tex]$$
\text{Change in speed} = 13.8375 - 6.58125 = 7.25625,
$$[/tex]
[tex]$$
\text{Time interval} = 8.2 - 3.9 = 4.3.
$$[/tex]
Now, compute the average:
[tex]$$
\text{Average rate of change}=\frac{7.25625}{4.3}\approx 1.69\ \text{feet per second}.
$$[/tex]
Thus, the estimated average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about [tex]$1.69$[/tex] feet per second.
[tex]$$
f(x)=1.6875\,x.
$$[/tex]
At [tex]$x=3.9$[/tex] seconds, the speed is
[tex]$$
f(3.9)=1.6875 \cdot 3.9=6.58125\ \text{feet per second}.
$$[/tex]
At [tex]$x=8.2$[/tex] seconds, the speed is
[tex]$$
f(8.2)=1.6875 \cdot 8.2=13.8375\ \text{feet per second}.
$$[/tex]
The average rate of change over the time interval is given by the formula
[tex]$$
\text{Average rate of change}=\frac{f(8.2)-f(3.9)}{8.2-3.9}.
$$[/tex]
Substitute the computed speeds:
[tex]$$
\text{Average rate of change}=\frac{13.8375-6.58125}{8.2-3.9}.
$$[/tex]
Calculate the differences:
[tex]$$
\text{Change in speed} = 13.8375 - 6.58125 = 7.25625,
$$[/tex]
[tex]$$
\text{Time interval} = 8.2 - 3.9 = 4.3.
$$[/tex]
Now, compute the average:
[tex]$$
\text{Average rate of change}=\frac{7.25625}{4.3}\approx 1.69\ \text{feet per second}.
$$[/tex]
Thus, the estimated average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about [tex]$1.69$[/tex] feet per second.
