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Let's analyze the given quadratic function, which models the height of the ball:

\[
y = -\frac{32}{(20)^2}x^2 + x + 3
\]

where:
- \(x\) is the horizontal distance traveled in feet,
- \(y\) is the height of the ball in feet.

### Part (a): Maximum Height of the Ball

The quadratic equation is in the standard form \(y = ax^2 + bx + c\), where \(a = -\frac{32}{400}\), \(b = 1\), and \(c = 3\). To find the maximum height, we need to find the vertex of the parabola, which occurs at:

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

Once we have \(x\), we substitute it back into the equation to find the maximum height \(y\).

### Part (b): Horizontal Distance When the Ball Hits the Ground

The ball hits the ground when \(y = 0\). We can solve the quadratic equation by setting \(y = 0\) and solving for \(x\). This will give the horizontal distance when the ball hits the ground.

Let me calculate both values for you.### Part (a): Maximum Height
The maximum height of the ball is attained at \( x = 6.25 \) feet. Substituting this value into the height equation gives:
\[
\text{Maximum height} = 6.125 \, \text{feet}.
\]

### Part (b): Horizontal Distance When the Ball Hits the Ground
The ball hits the ground at \( x = 15.0 \) feet. (We disregard the negative solution \( x = -2.5 \) as it does not make physical sense in this context.)

Do you need further details or explanations on these steps?

Here are five related questions for deeper understanding:
1. How would the trajectory change if the initial height were different?
2. What is the significance of the coefficient \(-\frac{32}{400}\) in this context?
3. How would you interpret the vertex form of a parabola in relation to projectile motion?
4. Can the trajectory be solved using a different method, like completing the square?
5. How would wind resistance alter the shape of the projectile's path?

**Tip**: The maximum or minimum point of a quadratic function is always at the vertex, and its x-coordinate is found using \(x = -\frac{b}{2a}\).

A ball is thrown from a height of 3 feet, and its path is modeled by the quadratic equation y = -(32/20^2)x^2 + x + 3. Find the maximum height and the horizontal distance when the ball hits the ground.

This problem involves finding the maximum height and horizontal distance of a ball thrown at an angle. The quadratic equation y = -(32/20^2)x^2 + x + 3 models the trajectory. Learn how to use the vertex formula and solve for when the ball hits the ground.

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