Answer :
To estimate the average rate of change of the elevator's speed function [tex]\( f(x) = 1.6875x \)[/tex] between 3.9 seconds and 8.2 seconds, follow the steps below:
1. Find the speed at 3.9 seconds:
Use the function [tex]\( f(x) = 1.6875x \)[/tex] to calculate the speed when [tex]\( x = 3.9 \)[/tex]:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58 \, \text{feet/second (approximately)}
\][/tex]
2. Find the speed at 8.2 seconds:
Again, use the function to find the speed when [tex]\( x = 8.2 \)[/tex]:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.84 \, \text{feet/second (approximately)}
\][/tex]
3. Calculate the average rate of change:
The average rate of change of the function from 3.9 seconds to 8.2 seconds is given by the formula:
[tex]\[
\frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
Substitute the values found:
[tex]\[
\frac{13.84 - 6.58}{8.2 - 3.9} = \frac{7.26}{4.3} = 1.69 \, \text{feet/second (rounded to two decimal places)}
\][/tex]
Therefore, the estimated average rate of change between 3.9 seconds and 8.2 seconds is about 1.69 feet/second.
1. Find the speed at 3.9 seconds:
Use the function [tex]\( f(x) = 1.6875x \)[/tex] to calculate the speed when [tex]\( x = 3.9 \)[/tex]:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58 \, \text{feet/second (approximately)}
\][/tex]
2. Find the speed at 8.2 seconds:
Again, use the function to find the speed when [tex]\( x = 8.2 \)[/tex]:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.84 \, \text{feet/second (approximately)}
\][/tex]
3. Calculate the average rate of change:
The average rate of change of the function from 3.9 seconds to 8.2 seconds is given by the formula:
[tex]\[
\frac{f(8.2) - f(3.9)}{8.2 - 3.9}
\][/tex]
Substitute the values found:
[tex]\[
\frac{13.84 - 6.58}{8.2 - 3.9} = \frac{7.26}{4.3} = 1.69 \, \text{feet/second (rounded to two decimal places)}
\][/tex]
Therefore, the estimated average rate of change between 3.9 seconds and 8.2 seconds is about 1.69 feet/second.
