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Answer :
We are given that the elevator's speed is modeled by the function
[tex]$$
f(x) = 1.6875x,
$$[/tex]
where [tex]$x$[/tex] represents time in seconds. We need to find the average rate of change between [tex]$x = 3.9$[/tex] seconds and [tex]$x = 4.0$[/tex] seconds.
The average rate of change of a function between two points [tex]$a$[/tex] and [tex]$b$[/tex] is given by
[tex]$$
\frac{f(b) - f(a)}{b - a}.
$$[/tex]
In this problem, [tex]$a = 3.9$[/tex] and [tex]$b = 4.0$[/tex].
1. First, compute the speed at [tex]$x = 3.9$[/tex] seconds:
[tex]$$
f(3.9) = 1.6875 \times 3.9 \approx 6.58125 \text{ feet per second}.
$$[/tex]
2. Next, compute the speed at [tex]$x = 4.0$[/tex] seconds:
[tex]$$
f(4.0) = 1.6875 \times 4.0 = 6.75 \text{ feet per second}.
$$[/tex]
3. Now, calculate the average rate of change:
[tex]$$
\text{Average rate} = \frac{f(4.0) - f(3.9)}{4.0 - 3.9} = \frac{6.75 - 6.58125}{0.1}.
$$[/tex]
4. Simplify the numerator:
[tex]$$
6.75 - 6.58125 = 0.16875.
$$[/tex]
5. Divide by the time difference:
[tex]$$
\frac{0.16875}{0.1} = 1.6875 \text{ feet per second}.
$$[/tex]
6. Rounding this value to two decimal places gives:
[tex]$$
1.69 \text{ feet per second}.
$$[/tex]
Thus, the estimated average rate of change between [tex]$3.9$[/tex] seconds and [tex]$4.0$[/tex] seconds is about [tex]$1.69$[/tex] feet per second.
[tex]$$
f(x) = 1.6875x,
$$[/tex]
where [tex]$x$[/tex] represents time in seconds. We need to find the average rate of change between [tex]$x = 3.9$[/tex] seconds and [tex]$x = 4.0$[/tex] seconds.
The average rate of change of a function between two points [tex]$a$[/tex] and [tex]$b$[/tex] is given by
[tex]$$
\frac{f(b) - f(a)}{b - a}.
$$[/tex]
In this problem, [tex]$a = 3.9$[/tex] and [tex]$b = 4.0$[/tex].
1. First, compute the speed at [tex]$x = 3.9$[/tex] seconds:
[tex]$$
f(3.9) = 1.6875 \times 3.9 \approx 6.58125 \text{ feet per second}.
$$[/tex]
2. Next, compute the speed at [tex]$x = 4.0$[/tex] seconds:
[tex]$$
f(4.0) = 1.6875 \times 4.0 = 6.75 \text{ feet per second}.
$$[/tex]
3. Now, calculate the average rate of change:
[tex]$$
\text{Average rate} = \frac{f(4.0) - f(3.9)}{4.0 - 3.9} = \frac{6.75 - 6.58125}{0.1}.
$$[/tex]
4. Simplify the numerator:
[tex]$$
6.75 - 6.58125 = 0.16875.
$$[/tex]
5. Divide by the time difference:
[tex]$$
\frac{0.16875}{0.1} = 1.6875 \text{ feet per second}.
$$[/tex]
6. Rounding this value to two decimal places gives:
[tex]$$
1.69 \text{ feet per second}.
$$[/tex]
Thus, the estimated average rate of change between [tex]$3.9$[/tex] seconds and [tex]$4.0$[/tex] seconds is about [tex]$1.69$[/tex] feet per second.
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