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.

Answer 1

To find the average rate of change of the function \( f(x) = 1.6875x \) between \( x = 3.9 \) seconds and \( x = 8.2 \) seconds, we can use the formula for 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 evaluate \( f(3.9) \) and \( f(8.2) \):

1. Calculate \( f(3.9) \):
\[
f(3.9) = 1.6875 \times 3.9 = 6.57875
\]

2. Calculate \( f(8.2) \):
\[
f(8.2) = 1.6875 \times 8.2 = 13.8275
\]

Now we can plug these values into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(8.2) - f(3.9)}{8.2 - 3.9} = \frac{13.8275 - 6.57875}{8.2 - 3.9}
\]

Calculating the numerator:

\[
13.8275 - 6.57875 = 7.24875
\]

Calculating the denominator:

\[
8.2 - 3.9 = 4.3
\]

Now we can find the average rate of change:

\[
\text{Average Rate of Change} = \frac{7.24875}{4.3} \approx 1.6847
\]

Finally, rounding to two decimal places:

\[
\text{Average Rate of Change} \approx 1.68 \text{ feet per second}
\]

Thus, the average rate of change between 3.9 seconds and 8.2 seconds is approximately **1.68 feet per second**.