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Part 1

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 equals negative 0.02 left parenthesis x minus 18 right parenthesis squared plus 12y=−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? Explain.

**b.** Describe how you would find the​ ball's height when it crosses the net at 

xequals=30.

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Part 1

**a.** How far away is the ball from Amaya when it is at its maximum​ height? Explain.

The ball is 

enter your response here 

ft away from Amaya when it is at its maximum height. Use the 

▼
 
x-coordinate

y-coordinate
 
of the vertex.

The ball isft away from Amaya when it is at its maximum height.Use theof the vertex.

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The equation given is \( y = -0.02(x - 18)^2 + 12 \), which represents the path of the ball. This is a quadratic equation in the form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

### **Part (a): Maximum Height**

The maximum height of the ball occurs at the vertex of the parabola because the parabola opens downwards (since \( a = -0.02 \), which is less than 0).

For the equation \( y = -0.02(x - 18)^2 + 12 \):
- The vertex occurs at \( x = 18 \).

Thus, the ball is 18 feet away from Amaya when it is at its maximum height.

**Final Answer (a):** The ball is 18 feet away from Amaya when it is at its maximum height. This is determined by using the **x-coordinate** of the vertex.

### **Part (b): Height of the Ball at the Net**

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

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

Calculate this to determine the height of the ball at that point.

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Would you like to proceed with the calculation for part (b) or have any other questions? Here are some related questions to consider:

1. What is the maximum height reached by the ball?
2. At what point does the ball hit the ground?
3. How does changing the coefficient \( a \) affect the trajectory of the ball?
4. How far does the ball travel horizontally before hitting the ground?
5. If Amaya moves closer to the net, how would the equation change?

**Tip:** The vertex of a parabola gives you key information such as the maximum or minimum value, depending on whether the parabola opens upwards or downwards.

Explore how to analyze the trajectory of a volleyball using quadratic equations. Find out the maximum height and height at net crossing using specific examples and calculations. Suitable for high school students.

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