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The problem asks you to determine the time when the ball reaches its maximum height and to find that maximum height. The height \( h(t) \) of the ball is given by the quadratic function:

\[
h(t) = -16t^2 + 32t + 4
\]

Here:
- \( t \) represents time in seconds,
- \( h(t) \) represents the height in feet.

To solve for the maximum height and the time at which it occurs, we use the properties of quadratic functions. The general form of a quadratic function is:

\[
h(t) = at^2 + bt + c
\]

For a quadratic function, the maximum (or minimum) occurs at the vertex. The time at which the maximum height is reached can be found using the formula for the vertex:

\[
t = \frac{-b}{2a}
\]

In this equation:
- \( a = -16 \)
- \( b = 32 \)

Substituting into the formula:

\[
t = \frac{-32}{2(-16)} = \frac{-32}{-32} = 1 \, \text{second}
\]

So, the ball reaches its maximum height at \( t = 1 \) second.

To find the maximum height, substitute \( t = 1 \) back into the height equation:

\[
h(1) = -16(1)^2 + 32(1) + 4
\]
\[
h(1) = -16 + 32 + 4 = 20 \, \text{feet}
\]

Thus, the maximum height of the ball is **20 feet**, and it occurs at **1 second**.

Would you like further details or explanations on any of the steps? Here are some related questions you might find useful:

1. How do you find the vertex of a quadratic function algebraically?
2. What is the significance of the coefficients \( a \), \( b \), and \( c \) in the quadratic formula?
3. How does the quadratic function describe the motion of a projectile?
4. What happens when \( a \) is positive versus when \( a \) is negative in a quadratic function?
5. How would you calculate the initial velocity from this function?

**Tip:** The formula for the vertex of a parabola, \( t = \frac{-b}{2a} \), is essential in many applications of quadratic equations, especially in physics for projectile motion problems!

A ball is thrown directly upward from a height of 4 feet, with an initial velocity of 32 feet per second. The function h(t) = -16t^2 + 32t + 4 gives the height of the ball, in feet, t seconds after it is thrown. Determine the time when the ball reaches its maximum height and find the maximum height.

Determine when a ball reaches its maximum height and find that height using the quadratic function h(t) = -16t^2 + 32t + 4. Learn how to apply the vertex formula for quadratic functions in this physics-based algebra problem, suitable for Grades 9-11.

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