To estimate the average rate of change of the function \( f(x) = 1.6875x \) between \( x = 3.9 \) seconds and \( x = 8.2 \) seconds, we will use the formula for average rate of change, which is given by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 3.9 \) and \( b = 8.2 \).
First, we calculate \( f(3.9) \) and \( f(8.2) \):
\[ f(3.9) = 1.6875 \times 3.9 = 6.58275 \text{ feet/second} \]
\[ f(8.2) = 1.6875 \times 8.2 = 13.8375 \text{ feet/second} \]
Next, we substitute these values into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{13.8375 - 6.58275}{8.2 - 3.9} \]
Calculating the difference in \( f \):
\[ 13.8375 - 6.58275 = 7.25475 \]
Calculating the difference in \( x \):
\[ 8.2 - 3.9 = 4.3 \]
Now we can calculate the average rate of change:
\[ \text{Average Rate of Change} = \frac{7.25475}{4.3} \approx 1.69 \text{ feet/second} \]
Rounding to two decimal places, the average rate of change is approximately:
\[ \text{Average Rate of Change} \approx 1.69 \text{ feet/second} \]
So the correct response is:
about 1.69 feet/second
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