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.

height of ball over time

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

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.

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.

height of ball over time

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

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.

Accepted Answer

# Brief Explanations:
- Analyze each option:
  - Option 1: At $ t = 0 $, $ h(t)=0 $, so initial height is 0, not 96. Eliminate.
  - Option 2: At $ t = 5 $, $ h(t)=0 $ (hits ground at 5s), not between 2 - 3s. Eliminate.
  - Option 3: Height at $ t = 2 $ and $ t = 3 $ is 96, and it's symmetric (e.g., $ t = 1 $ and $ t = 4 $ both 64, $ t = 0 $ and $ t = 5 $ both 0). The vertex (maximum) is at the midpoint of $ t = 2 $ and $ t = 3 $, but the height at those times is 96, and since the parabola opens downward, the maximum height is at least 96, but actually, since it's symmetric around $ t = 2.5 $, and the height at $ t = 2 $ and $ t = 3 $ is 96, the maximum height is 96 (because the parabola's vertex height is equal to the height at $ t = 2 $ and $ t = 3 $ as it's symmetric there). Wait, no—wait, the data: $ t = 0 $: 0, $ t = 1 $: 64, $ t = 2 $: 96, $ t = 3 $: 96, $ t = 4 $: 64, $ t = 5 $: 0. So the function is symmetric around $ t = 2.5 $. The height at $ t = 2 $ and $ t = 3 $ is 96, and since it's a parabola (projectile motion), the maximum height is at $ t = 2.5 $, and since the height at $ t = 2 $ and $ t = 3 $ is 96, and the parabola opens downward, the maximum height is 96 (because the vertex is the peak, and the points around it (t=2, t=3) have the same height, so the vertex height is 96). Wait, no—actually, if at t=2 and t=3, height is 96, and it's symmetric around t=2.5, then the vertex is at t=2.5, and the height there is the maximum. But since t=2 and t=3 are equidistant from 2.5, their heights are equal, so the maximum height is 96? Wait, no, maybe the maximum is higher? Wait, no, the data shows that at t=2 and t=3, height is 96, and at t=1 and t=4, it's 64, t=0 and t=5, 0. So the parabola is $ h(t)=-16t(t - 5) $ (since roots at t=0 and t=5). Let's check: $ h(1)=-16(1)(-4)=64 $, correct. $ h(2)=-16(2)(-3)=96 $, correct. $ h(3)=-16(3)(-2)=96 $, correct. $ h(4)=-16(4)(-1)=64 $, correct. So the equation is $ h(t)=-16t^2 + 80t $. Let's compute $ h(2.5) $: $ -16(6.25) + 80(2.5)= -100 + 200 = 100 $? Wait, wait, that's a mistake. Wait, $ h(t)=-16t(t - 5)=-16t^2 + 80t $. Then $ h(2.5)=-16(6.25) + 80(2.5)=-100 + 200 = 100 $. Wait, but the table says at t=2 and t=3, h(t)=96. Wait, that's a contradiction. Wait, maybe the table is approximate? Or maybe I miscalculated. Wait, $ h(2)=-16(4) + 80(2)=-64 + 160 = 96 $, correct. $ h(3)=-16(9) + 80(3)=-144 + 240 = 96 $, correct. $ h(2.5)=-16(6.25) + 80(2.5)=-100 + 200 = 100 $. Oh! So the table's t=2 and t=3 have h(t)=96, but the actual maximum at t=2.5 is 100. Wait, but the options: "The maximum height of the ball must be 96 feet." Is that true? No, because at t=2.5, it's 100. Wait, but maybe the table is the data, so maybe the model is different. Wait, the problem is about the statements. Let's re-examine each option:

1. Initial height: t=0, h=0. So initial height is 0, not 96. Eliminate.

2. Ball hits ground between 2 and 3? No, at t=5, h=0, so it hits at 5, not between 2-3. Eliminate.

3. Maximum height must be 96? Wait, but according to the quadratic model, it's 100. But maybe the table is the data, and since at t=2 and t=3, height is 96, and it's symmetric, the maximum is at t=2.5, and since the height at t=2 and t=3 is 96, and the parabola is symmetric, the maximum height is 96? Wait, no—if the height at t=2 and t=3 is 96, and it's increasing then decreasing, then the maximum is between t=2 and t=3, so the height there is higher than 96? But the table says t=2:96, t=3:96. So maybe the data is such that the maximum height is 96, because at t=2 and t=3, it's 96, and before t=2, it was increasing (from 0 at t=0, 64 at t=1, 96 at t=2), then at t=3, still 96, then decreasing (64 at t=4, 0 at t=5). So the height increases from t=0 to t=2, stays at 96 from t=2 to t=3, then decreases. So the maximum height is 96. Wait, that could be the case if the velocity is zero between t=2 and t=3, but projectile motion is a parabola, so velocity is zero at the vertex. But the table shows constant height from t=2 to t=3, which would mean zero velocity there, so the maximum height is 96. So maybe the model is a piecewise function, but the options: "The maximum height of the ball must be 96 feet." Let's check the other option: "The maximum height of the ball was reached 2.5 seconds after it was thrown." Since the data is symmetric around t=2.5 (t=0 and t=5, t=1 and t=4, t=2 and t=3), the vertex (maximum) is at t=2.5. So that statement is true. Wait, but earlier I thought the height at t=2.5 is 100, but maybe the table's data is such that the height at t=2 and t=3 is 96, and it's symmetric, so the maximum is at t=2.5, regardless of the height. Wait, the question is which statement is true. Let's re-express:

- Option 1: Initial height (t=0) is 0, not 96. False.

- Option 2: Hits ground between 2-3? No, hits at 5. False.

- Option 3: Maximum height must be 96? If the height at t=2 and t=3 is 96, and it's symmetric, then the maximum is at t=2.5, and since the height at t=2 and t=3 is 96, and it's increasing before t=2 and decreasing after t=3, the maximum height is 96 (because it reaches 96 at t=2, stays there until t=3, then decreases). So maybe that's true. Wait, but if it's a parabola, it should be symmetric, so t=2.5 is the vertex, and the height there is the maximum. But the table says t=2:96, t=3:96, so the height at t=2.5 is 96? No, that would mean the parabola has a flat top, which is not possible in projectile motion, but the problem is about the data given, not the physics model. So based on the table:

From t=0 to t=2: height increases (0→64→96).

From t=2 to t=3: height stays at 96.

From t=3 to t=5: height decreases (96→64→0).

So the maximum height is 96, reached between t=2 and t=3 (or at t=2 and t=3). Then the statement "The maximum height of the ball must be 96 feet" is true? Wait, but the other option: "The maximum height of the ball was reached 2.5 seconds after it was thrown." If the height is constant from t=2 to t=3, then the maximum is reached at any time between t=2 and t=3, including 2.5. But does that mean the maximum height was reached at 2.5? If the height is 96 from t=2 to t=3, then the maximum height is 96, and it was reached at any time in [2,3], including 2.5. But let's check the options again:

Wait, the four options:

1. Initial height 96: false (t=0:0).

2. Hits ground between 2-3: false (t=5:0).

3. Maximum height must be 96: let's see, at t=2 and t=3, height is 96, and before t=2, it was increasing (so at t=2, it's 96, then at t=3, still 96, then decreasing. So the maximum height is 96. So this could be true.

4. Maximum height reached at 2.5: if the height is constant from t=2 to t=3, then the maximum is reached at any time in that interval, including 2.5, but does that mean "was reached 2.5 seconds after"? If the height is 96 from t=2 to t=3, then the maximum height is 96, and it was reached at 2.5 seconds (the midpoint), but the height at 2.5 is 96 (same as t=2 and t=3). So both options 3 and 4? Wait, no—wait the table:

t | h(t)

0 | 0

1 | 64

2 | 96

3 | 96

4 | 64

5 | 0

So the function is symmetric around t=2.5. So the vertex (maximum) is at t=2.5, and the height there is the maximum. Now, what's h(2.5)? Let's calculate using symmetry. The difference between t=2 and t=2.5 is 0.5, and between t=2.5 and t=3 is 0.5. The height at t=1 (2.5 - 1.5) is 64, at t=4 (2.5 + 1.5) is 64. At t=0 (2.5 - 2.5) is 0, at t=5 (2.5 + 2.5) is 0. At t=2 (2.5 - 0.5) is 96, at t=3 (2.5 + 0.5) is 96. So the height at t=2.5 should be the maximum. Let's model it as a quadratic function: h(t) = at² + bt + c. We know h(0)=0 ⇒ c=0. h(1)=64 ⇒ a + b = 64. h(2)=96 ⇒ 4a + 2b = 96. Solve: from first equation, b=64 - a. Substitute into second: 4a + 2(64 - a)=96 ⇒ 4a + 128 - 2a=96 ⇒ 2a= -32 ⇒ a= -16. Then b=64 - (-16)=80. So h(t)= -16t² + 80t. Then h(2.5)= -16(6.25) + 80(2.5)= -100 + 200=100. Oh! So the height at t=2.5 is 100, which is higher than 96. So the table's data at t=2 and t=3 is 96, but the actual maximum height is 100 at t=2.5. So that means:

- Option 3: "The maximum height of the ball must be 96 feet." is false, because the maximum is 100.

- Option 4: "The maximum height of the ball was reached 2.5 seconds after it was thrown." is true, because at t=2.5, the height is 100 (maximum), so the maximum height was reached at 2.5 seconds.

Wait, but the table says h(2)=96 and h(3)=96, but according to the quadratic model, h(2)= -16(4) + 80(2)= -64 + 160=96, correct. h(3)= -16(9) + 80(3)= -144 + 240=96, correct. h(2.5)=100, which is not in the table, but the table is just sample data. So the key is:

- Initial height: t=0, h=0 → false.

- Hits ground between 2-3: t=5, h=0 → false.

- Maximum height must be 96: no, because at t=2.5, it's 100 → false.

- Maximum height reached at 2.5 seconds: yes, because the vertex (maximum) of the parabola (projectile motion) is at t= -b/(2a)= -80/(2*(-16))= 2.5, so the maximum height is reached at 2.5 seconds.

Ah! So that's the key. The projectile motion is a quadratic function (parabola) opening downward, so the vertex (maximum) is at t= -b/(2a). From the data, we can find the equation, and the vertex is at t=2.5. So the statement "The maximum height of the ball was reached 2.5 seconds after it was thrown" is true.

Wait, but the height at t=2.5 is 100, which is higher than 96, but the statement is about when the maximum height was reached, not the height. So even though the table doesn't show h(2.5), the symmetry (t=0 and t=5, t=1 and t=4, t=2 and t=3) implies the vertex is at t=2.5, so the maximum height was reached at 2.5 seconds.

So the correct statement is the fourth option: "The maximum height of the ball was reached 2.5 seconds after it was thrown."

Wait, but let's confirm with the options:

1. Initial height 96: false (t=0:0).

2. Hits ground between 2-3: false (t=5:0).

3. Maximum height must be 96: false (as h(2.5)=100).

4. Maximum height reached at 2.5: true, because the vertex (maximum) of the parabola is at t=2.5 (due to symmetry), so the maximum height occurs at 2.5 seconds.

Yes, that makes sense. The symmetry of the data (t and 5 - t have the same h(t)) shows the vertex is at t=2.5, so the maximum height is reached at 2.5 seconds.
# Answer:
The maximum height of the ball was reached 2.5 seconds after it was thrown.

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