Question

a ball is thrown straight up into the air. the table shows the data collected over t seconds, where h(t) is the height of the ball, in feet. which statement is true? the initial height of the ball is 96 feet. the ball will hit the ground between 2 and 3 seconds after it was thrown. the maximum height of the ball must be 96 feet. the maximum height of the ball was reached 2.5 seconds after it was thrown. height of ball over time

t	h(t)
1	64
2	96
3	96
4	64
5	0

Explanation:

Step1: Check initial height

When \(t = 0\), \(h(0)=0\), so the initial height is 0 feet, not 96 feet.

Step2: Check when ball hits ground

The ball hits the ground when \(h(t)=0\). From the table, \(h(0) = 0\) (initial - start) and \(h(5)=0\). It hits the ground at \(t = 5\) seconds, not between 2 and 3 seconds.

Step3: Check maximum height

The height values are \(h(0)=0\), \(h(1) = 64\), \(h(2)=96\), \(h(3)=96\), \(h(4)=64\), \(h(5)=0\). The maximum height is 96 feet, but we can't be sure it's the absolute maximum without more data (it could be higher between 2 - 3 seconds).

Step4: Check time of maximum height

We don't have enough data to say the maximum height was reached at \(t = 2.5\) seconds. Just because \(h(2)=h(3)=96\) doesn't mean the max was at 2.5 seconds. But the maximum height of the ball must be at least 96 feet.

Answer:

The maximum height of the ball must be 96 feet.