Answer
Between 0 and 0.5 seconds
Work Step by Step
Given: The position function
s (t) = -16t^2+ v0 t+ s0
Here, v0 = 8 feet per second and s0 = 87 feet
Explanation:
In order to find the period of time during which the ball’s height exceeds that of the cliff, let us solve the function
s (t) = -16t^2+ v0 t+ s0
On substituting the given data, we have
s (t) = -16t^2+ 8 t+ 87
Here are the steps required for Solving Polynomial Inequalities:
Step 1: One side must be zero and the other side can have only one fraction, so we simplify the fractions if there is more than one fraction.
Thus,
-16t^2+ 8t > 0
Step 2: Critical or Key Values are first evaluated. In order to this, set the equation equal to zero and then simplified inequality is solved.
-16t^2+ 8t = 0
-8 t (2t-1) = 0
-8 t = 0 or 2t-1 = 0
This implies
t = 0 or t = 0.5
Step 3: Locate the boundary points on a number line found in Step 2 to divide the number line into intervals. The boundary points are shown as follows:
The boundary points divide the number line into three intervals:
(-infinity, 0), (0, 0.5), (1.5, infinity)
For our purposes the mathematical model is useful only from t = 0 until the diver hits the ground.
Let us determine the time when the diver hits the ground.
s (t) = 0
-16t^2+ v0 t+ s0 = 0
-16t^2+ 8 t+ 87 = 0
This implies
4t-1 = 9.38
t ≈ 2.6
As t ≥ 0only t = 2.6 fits. Therefore, we use the intervals (0, 0.5), ( 0.5, 2.6).
Step 4: Now, one test value within each interval is chosen and f is evaluated at that number.
Interval Test value Substitute into
f(t) = - 16t^2 + 8 t Conclusion
(0, 0.5) 0.25 f(0.25) = - 16(0.25)^2 + 8(0.25) f(t) > 0
= 1, Positive
(0, 2.6) 1 f(0.25) = - 16(0.25)^2 + 8(0.25)
= - 8, Negative f(t) < 0
Step 5: Write the solution set, selecting the interval or intervals that satisfy the given inequality
f (t) > 0 [f(t) = - 16t^2 + 8 t]
Based on our work done in Step 4, we see that f (t) > 0 for all x in (0, 0.5).
Conclusion: This means that the diver’s height exceeds that of the cliff between 0 and 0.5 second.