Answer :
To estimate the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds, follow these steps:
1. Understand the Function:
The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] is the time in seconds.
2. Calculate the Speed at [tex]\( x = 3.9 \)[/tex] seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58 \text{ feet per second}
\][/tex]
3. Calculate the Speed at [tex]\( x = 8.2 \)[/tex] seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.84 \text{ feet per second}
\][/tex]
4. Determine the Average Rate of Change:
The average rate of change for a function between two points [tex]\( x = a \)[/tex] and [tex]\( x = b \)[/tex] is calculated as:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
Substitute the given values:
[tex]\[
\frac{13.84 - 6.58}{8.2 - 3.9} = \frac{7.26}{4.3} \approx 1.69
\][/tex]
5. Conclusion:
The average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is approximately [tex]\( 1.69 \)[/tex] feet per second.
Thus, the correct choice is about 1.69 feet/second.
1. Understand the Function:
The speed of the elevator is given by the function [tex]\( f(x) = 1.6875x \)[/tex], where [tex]\( x \)[/tex] is the time in seconds.
2. Calculate the Speed at [tex]\( x = 3.9 \)[/tex] seconds:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58 \text{ feet per second}
\][/tex]
3. Calculate the Speed at [tex]\( x = 8.2 \)[/tex] seconds:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.84 \text{ feet per second}
\][/tex]
4. Determine the Average Rate of Change:
The average rate of change for a function between two points [tex]\( x = a \)[/tex] and [tex]\( x = b \)[/tex] is calculated as:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
Substitute the given values:
[tex]\[
\frac{13.84 - 6.58}{8.2 - 3.9} = \frac{7.26}{4.3} \approx 1.69
\][/tex]
5. Conclusion:
The average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is approximately [tex]\( 1.69 \)[/tex] feet per second.
Thus, the correct choice is about 1.69 feet/second.
