Question

The speed of an elevator (in feet per second) is modeled by the function f(x)=1.6875x , where x is time in seconds. Estimate the average rate of change between 3.9 seconds and 8.2 seconds. Round the final answer to two decimal places.(1 point) Responses about 4.00 feet/second about 4.00 feet/second about 1.69 feet/second about 1.69 feet/second about 0.59 feet/second about 0.59 feet/second about 6.75 feet/second

Answer 1

To find the average rate of change of the elevator's speed between \(x = 3.9\) seconds and \(x = 8.2\) seconds, we can use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \(a = 3.9\) and \(b = 8.2\).

First, we need to calculate \(f(3.9)\) and \(f(8.2)\):

1. **Calculate \(f(3.9)\)**:
\[
f(3.9) = 1.6875 \times 3.9 = 6.58575 \text{ feet/second}
\]

2. **Calculate \(f(8.2)\)**:
\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet/second}
\]

Now, we find the average rate of change:
\[
\text{Average Rate of Change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9} = \frac{13.8375 - 6.58575}{8.2 - 3.9}
\]
Calculating the numerator:
\[
13.8375 - 6.58575 = 7.25175
\]
Calculating the denominator:
\[
8.2 - 3.9 = 4.3
\]
Now we can calculate the average rate of change:
\[
\text{Average Rate of Change} = \frac{7.25175}{4.3} \approx 1.6883 \text{ feet/second}
\]

Rounding to two decimal places gives:
\[
\text{Average Rate of Change} \approx 1.69 \text{ feet/second}
\]

The final answer is **about 1.69 feet/second**.