Let’s analyze and solve the given quadratic function step by step.

The height of the baseball is modeled by:
$h(t) = -6(t - 2.5)^2 + 38.5$

a) Maximum Height of the Ball

The function is written in vertex form:
$h(t) = a(t - h)^2 + k$
Here, the vertex of the parabola $(h, k)$ represents the maximum point since the coefficient $a = -6$ is negative (downward opening parabola).

• From the equation, $h = 2.5$ and $k = 38.5$.
• The maximum height is $k = 38.5$ m.

Answer (a): The maximum height of the ball is 38.5 m.

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b) Height of the Ball When It Was Hit

To find the height of the ball when it was hit ($t = 0$), substitute $t = 0$ into the equation: $h(0) = -6(0 - 2.5)^2 + 38.5$ $h(0) = -6(2.5)^2 + 38.5 = -6(6.25) + 38.5 = -37.5 + 38.5 = 1$

Answer (b): The height of the ball when it was hit is 1 m.

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c) Time for the Ball to Hit the Ground

The ball hits the ground when $h(t) = 0$. Solve for $t$: $-6(t - 2.5)^2 + 38.5 = 0$ $-6(t - 2.5)^2 = -38.5 \quad \Rightarrow \quad (t - 2.5)^2 = \frac{38.5}{6} \quad \Rightarrow \quad (t - 2.5)^2 = 6.4167$ Take the square root: $t - 2.5 = \pm \sqrt{6.4167} \quad \Rightarrow \quad t - 2.5 = \pm 2.534$ $t = 2.5 + 2.534 \quad \text{or} \quad t = 2.5 - 2.534$ $t = 5.034 \quad \text{or} \quad t = -0.034$ Since time cannot be negative, the ball hits the ground at $t = 5.034$ seconds.

To the nearest second, 5 seconds.

Answer (c): The ball hits the ground 5 seconds after being hit.

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d) Height of the Ball 1 Second After Being Hit

Substitute $t = 1$ into the equation: $h(1) = -6(1 - 2.5)^2 + 38.5$ $h(1) = -6(-1.5)^2 + 38.5 = -6(2.25) + 38.5 = -13.5 + 38.5 = 25$

Answer (d): The height of the ball 1 second after it was hit is 25 m.

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e) Time Interval When the Ball’s Height is Greater Than 25 m

Solve for $t$ when $h(t) > 25$: $-6(t - 2.5)^2 + 38.5 > 25$ $-6(t - 2.5)^2 > -13.5 \quad \Rightarrow \quad (t - 2.5)^2 < \frac{13.5}{6} \quad \Rightarrow \quad (t - 2.5)^2 < 2.25$ Take the square root: $t - 2.5 < \sqrt{2.25} \quad \text{and} \quad t - 2.5 > -\sqrt{2.25}$ $t - 2.5 < 1.5 \quad \text{and} \quad t - 2.5 > -1.5$ $1 < t < 4$

Answer (e): The height of the ball is greater than 25 m during the time interval $1 < t < 4$ seconds. Accurate to one decimal place: (1.0, 4.0).

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Final Answers:

a) 38.5 m
b) 1 m
c) 5 seconds
d) 25 m
e) (1.0, 4.0) seconds

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Would you like a detailed explanation of any of the steps?
Here are 5 related questions for further exploration:

1. How would the equation change if the ball were hit from a higher initial height?
2. What is the significance of the coefficient $-6$ in the equation?
3. How would you determine the velocity of the ball at any given time?
4. What is the total distance traveled by the ball?
5. Can you express the function in standard quadratic form and find its roots?

Tip: Use graphing tools to visualize parabolas for a better understanding of their behavior!