Evaluate s(3) and s(3.25). The average velocity of the stone over the time interval [3,3.25] is.
Understand the Problem
The question is asking to evaluate the height of a stone at two specific times (3 seconds and 3.25 seconds) and then calculate the average velocity over the interval from 3 to 3.25 seconds. The provided heights are 56 feet and 66 feet respectively at those times.
Answer
The average velocity of the stone over the time interval $[3,3.25]$ is $40 \text{ feet per second}$.
Answer for screen readers
The average velocity of the stone over the time interval $[3,3.25]$ is $40 \text{ feet per second}$.
Steps to Solve
- Identify the heights at given times
We already have the heights:
- At $t = 3$ seconds, $s(3) = 56$ feet.
- At $t = 3.25$ seconds, $s(3.25) = 66$ feet.
- Calculate the average velocity
The average velocity over the interval $[3, 3.25]$ can be calculated using the formula:
$$ \text{Average Velocity} = \frac{s(3.25) - s(3)}{3.25 - 3} $$
- Substitute the known values into the formula
Substituting the values:
$$ \text{Average Velocity} = \frac{66 - 56}{3.25 - 3} $$
This simplifies to:
$$ \text{Average Velocity} = \frac{10}{0.25} $$
- Calculate the average velocity
Now calculate the average velocity:
$$ \text{Average Velocity} = 40 \text{ feet per second} $$
The average velocity of the stone over the time interval $[3,3.25]$ is $40 \text{ feet per second}$.
More Information
Average velocity measures how fast something is moving over a specified interval. In this case, it indicates the speed of the stone as it is thrown upwards.
Tips
- Forgetting to convert time differences properly, such as mistaking 0.25 seconds for 1 second.
- Incorrectly substituting values into the average velocity formula.
- Not simplifying the fraction completely.
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