Answer :
To estimate the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds, we can use the function provided, [tex]\( f(x) = 1.6875x \)[/tex], which models the speed of the elevator at any given time [tex]\( x \)[/tex].
Here’s a step-by-step guide to solving the problem:
1. Calculate the speed at 3.9 seconds:
- Use the function [tex]\( f(x) = 1.6875x \)[/tex].
- Substitute [tex]\( x = 3.9 \)[/tex] into the function:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \, \text{feet per second}
\][/tex]
2. Calculate the speed at 8.2 seconds:
- Again, use the function [tex]\( f(x) = 1.6875x \)[/tex].
- Substitute [tex]\( x = 8.2 \)[/tex] into the function:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \, \text{feet per second}
\][/tex]
3. Find the average rate of change between these two times:
- The average rate of change of a function between two points is given by the formula:
[tex]\[
\frac{f(t_2) - f(t_1)}{t_2 - t_1}
\][/tex]
- For this example, [tex]\( t_1 = 3.9 \)[/tex] and [tex]\( t_2 = 8.2 \)[/tex]:
[tex]\[
\frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} \approx 1.69
\][/tex]
4. Round to two decimal places:
- The final average rate of change, rounded to two decimal places, is approximately 1.69 feet per second.
Therefore, the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about 1.69 feet per second.
Here’s a step-by-step guide to solving the problem:
1. Calculate the speed at 3.9 seconds:
- Use the function [tex]\( f(x) = 1.6875x \)[/tex].
- Substitute [tex]\( x = 3.9 \)[/tex] into the function:
[tex]\[
f(3.9) = 1.6875 \times 3.9 = 6.58125 \, \text{feet per second}
\][/tex]
2. Calculate the speed at 8.2 seconds:
- Again, use the function [tex]\( f(x) = 1.6875x \)[/tex].
- Substitute [tex]\( x = 8.2 \)[/tex] into the function:
[tex]\[
f(8.2) = 1.6875 \times 8.2 = 13.8375 \, \text{feet per second}
\][/tex]
3. Find the average rate of change between these two times:
- The average rate of change of a function between two points is given by the formula:
[tex]\[
\frac{f(t_2) - f(t_1)}{t_2 - t_1}
\][/tex]
- For this example, [tex]\( t_1 = 3.9 \)[/tex] and [tex]\( t_2 = 8.2 \)[/tex]:
[tex]\[
\frac{13.8375 - 6.58125}{8.2 - 3.9} = \frac{7.25625}{4.3} \approx 1.69
\][/tex]
4. Round to two decimal places:
- The final average rate of change, rounded to two decimal places, is approximately 1.69 feet per second.
Therefore, the average rate of change of the elevator's speed between 3.9 seconds and 8.2 seconds is about 1.69 feet per second.
