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Let's work through part **a** of the problem using the given quadratic equation for the ball's path.

The equation for the path of the ball is given as:
\[
y = x - 0.04x^2
\]
Where:
- \( x \) is the horizontal distance (in meters),
- \( y \) is the vertical height (in meters).

### Part (a): Finding the horizontal distance the ball travels
The horizontal distance where the ball lands on the field occurs when the ball reaches the ground again. This happens when \( y = 0 \). So, we need to solve the equation:
\[
0 = x - 0.04x^2
\]
Rearranging this equation:
\[
0.04x^2 = x
\]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[
0.04x = 1
\]
Solving for \( x \):
\[
x = \frac{1}{0.04} = 25
\]

Thus, the horizontal distance the ball travels is **25 meters**.

Would you like to go over part **b** next? Here are some relative questions to explore further:

1. What is the general form of a quadratic function, and how does it relate to projectile motion?
2. How would you calculate the maximum height the ball reaches?
3. Can the equation for the ball's path be derived from physics principles like kinematic equations?
4. How can you verify the solution using a graphing method or a calculator?
5. How does changing the coefficient of \( x^2 \) affect the trajectory of the ball?

**Tip:** In projectile motion, the coefficient of the quadratic term affects the "sharpness" or "flatness" of the parabolic path. A smaller coefficient results in a wider arc.

Explore how to solve a quadratic function that models the projectile motion of a ball kicked into the air. Learn to find the horizontal distance, rewrite the function in vertex form, and calculate the maximum height reached by the ball. Suitable for high school math students.

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