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The problem involves Amaya serving a volleyball, with the ball's height modeled by the quadratic equation:

\[
y = -0.02(x - 18)^2 + 12
\]

Where:
- \(x\) is the horizontal distance (in feet) from Amaya.
- \(y\) is the height (in feet) of the ball.

Let’s analyze the two parts of the question:

### Part (a): How far is the ball from Amaya when it is at its maximum height? Explain.

The equation provided is in vertex form:
\[
y = a(x - h)^2 + k
\]
Where \( (h, k) \) is the vertex of the parabola. In this case:
- \(h = 18\), which represents the horizontal distance from Amaya where the ball reaches its maximum height.
- \(k = 12\), which represents the maximum height of the ball.

So, the ball is **18 feet** away from Amaya when it reaches its maximum height.

### Part (b): Describe how you would find the ball’s height when it crosses the net at \(x = 30\).

To find the height of the ball when it crosses the net, substitute \(x = 30\) into the equation for \(y\):

\[
y = -0.02(30 - 18)^2 + 12
\]

First, calculate \( (30 - 18)^2 \):
\[
(30 - 18) = 12
\]
\[
12^2 = 144
\]

Now substitute into the equation:
\[
y = -0.02(144) + 12
\]
\[
y = -2.88 + 12
\]
\[
y = 9.12
\]

Thus, the ball is approximately **9.12 feet** high when it crosses the net at \(x = 30\).

---

Would you like more details on any step? Here are 5 related questions for practice:

1. How do you determine the vertex of a quadratic equation in standard form?
2. What does the value of \(a\) in the equation \(y = a(x - h)^2 + k\) tell you about the parabola's direction?
3. How do you find the horizontal distance where the ball is at a specific height, such as 10 feet?
4. Can you explain why the maximum height occurs at \(x = h\) in vertex form?
5. How would the trajectory change if the value of \(a\) were less negative (e.g., \(a = -0.01\))?

### Tip: 
To quickly find the maximum or minimum of a parabola, use the vertex form of the equation—it directly gives the vertex!

Amaya is standing 30 ft from a volleyball net. The net is 8 ft high. Amaya serves the ball. The path of the ball is modeled by the equation y = -0.02(x - 18)^2 + 12, where x is the ball's horizontal distance in feet from Amaya's position and y is the distance in feet from the ground to the ball. a. How far away is the ball from Amaya when it is at its maximum height? b. Describe how you would find the ball’s height when it crosses the net at x = 30.

Learn how to calculate the maximum height and the height of a volleyball as it crosses the net using the quadratic equation y = -0.02(x - 18)^2 + 12. The solution demonstrates how to interpret vertex form and solve for specific values of x.

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