Evaluate s(3) and s(3.25).
Understand the Problem
The question is asking to evaluate the height of a stone at two specific times, 3 seconds and 3.25 seconds, using the provided height function s(t). Additionally, it requires calculating the average velocity of the stone over the time interval between these two points in time.
Answer
$$ s(3) = 56, \quad s(3.25) = 66, \quad \text{Average Velocity} = 40 \text{ feet per second} $$
Answer for screen readers
$$ s(3) = 56 $$
$$ s(3.25) = 66 $$
Average Velocity = 40 feet per second.
Steps to Solve
-
Evaluating s(3)
We know that after 3 seconds, the height of the stone is 56 feet.
Thus, we have:
$$ s(3) = 56 $$ -
Evaluating s(3.25)
After 3.25 seconds, the height of the stone is 66 feet.
Therefore:
$$ s(3.25) = 66 $$ -
Finding Average Velocity
The average velocity over the interval [3, 3.25] can be calculated using the formula:
$$ \text{Average Velocity} = \frac{s(b) - s(a)}{b - a} $$
Where ( a = 3 ) and ( b = 3.25 ).
Substituting the values:
$$ \text{Average Velocity} = \frac{s(3.25) - s(3)}{3.25 - 3} = \frac{66 - 56}{3.25 - 3} $$
Calculating the differences:
$$ \text{Average Velocity} = \frac{10}{0.25} = 40 \text{ feet per second} $$
$$ s(3) = 56 $$
$$ s(3.25) = 66 $$
Average Velocity = 40 feet per second.
More Information
The average velocity over the interval shows how fast the stone was rising between 3 and 3.25 seconds. In this instance, it rose an average of 40 feet per second, which indicates a fairly rapid ascent.
Tips
- Forgetting to simplify the fraction when calculating average velocity.
- Miscalculating the height values by confusing the time intervals.
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