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 \(f(8.2)\):
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Calculate \(f(3.9)\): \[ f(3.9) = 1.6875 \times 3.9 = 6.58575 \text{ feet/second} \]
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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.
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