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The exercise presents a situation where a volleyball player is serving a ball, with the goal being to analyze the trajectory of the ball.

Here’s a summary of the problem:

- **Diagram**: It shows a volleyball net at a height of 2.43 meters, with the player serving from point B (on the y-axis, 1 meter above ground). The net is 9 meters away from the player, and the total court length is 18 meters.

The ball's height \( h(x) \) (in meters) as a function of its horizontal position \( x \) (in meters) is modeled by the quadratic equation:
\[
h(x) = -0.05x^2 + 0.6x + 1
\]
### Questions to answer:
1. **Calculate the initial height** of the ball at the start of its trajectory (i.e., the height at \( x = 0 \)).
2. **Show that the ball passes over the net**, which is located at \( x = 9 \) meters, with a height of 2.43 meters.
3. **Determine if the ball lands inside the court** at point C, which is 18 meters away from the player (i.e., when \( x = 18 \)).

Let me solve these questions one by one.

### 1. Height of the ball at \( x = 0 \):
At the start of the trajectory, \( x = 0 \). Using the given equation \( h(x) = -0.05x^2 + 0.6x + 1 \), we can substitute \( x = 0 \) to find the initial height \( h(0) \).

Let me calculate that.At \( x = 0 \), the initial height of the ball is 1 meter, as expected from the given data.

### 2. Height of the ball at the net (\( x = 9 \)):

Next, let’s determine the ball’s height at the net, which is located 9 meters away from the player (\( x = 9 \)), and check if it exceeds the net height of 2.43 meters.

Let me calculate that.At \( x = 9 \) meters, the height of the ball is approximately 2.35 meters, which is slightly below the net height of 2.43 meters. Therefore, the ball does **not** pass over the net.

### 3. Height of the ball at \( x = 18 \) meters:

Lastly, let's calculate the height of the ball at \( x = 18 \), the other side of the court, to check if the ball lands inside the court.

Let me calculate that now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can manually calculate the height at \( x = 18 \) by substituting it into the equation:
\[
h(18) = -0.05(18)^2 + 0.6(18) + 1
\]

Would you like me to help further with that or explain any specific part of the solution so far?

Here are 5 related questions to deepen your understanding:
1. How would the trajectory change if the player increased their serve power?
2. Can you derive the vertex of the quadratic equation to determine the maximum height of the ball?
3. How would the equation change if the net height were reduced to 2 meters?
4. What is the interpretation of the quadratic coefficient \( -0.05 \)?
5. Can you calculate the horizontal distance at which the ball reaches its maximum height?

**Tip**: To check if the ball lands inside the court, find the roots of the quadratic equation to determine where the ball touches the ground.

A volleyball player practices serving. The trajectory of the ball is given by h(x) = -0.05x^2 + 0.6x + 1, where h(x) represents the height in meters and x the distance from the player in meters. The net is 9 meters away and 2.43 meters high. Tasks: 1. Calculate the height of the ball at x = 0. 2. Show that the ball passes over the net. 3. Determine if the ball lands inside the court, i.e., before or at x = 18 meters.

Analyze the trajectory of a volleyball serve using a quadratic function. Calculate the initial height, verify if the ball passes over the net at 9 meters, and determine if it lands inside the court. Suitable for Grades 9-10.

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