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The problem provided models the height of a ball, \( f(x) = -0.2x^2 + 2.1x + 8 \), where \( x \) represents the ball's horizontal distance in feet from the point where it was thrown.

You are asked to find:
1. **(a)** The maximum height of the ball and how far from the point of release this occurs, which has already been calculated:
   - Maximum height = 13.5 feet
   - Occurs at a horizontal distance of 5.3 feet from the release point.

2. **(b)** The total horizontal distance the ball travels before hitting the ground.

### Part (b): How far does the ball travel horizontally before hitting the ground?

To find this, we need to determine when the ball hits the ground. This happens when the height \( f(x) = 0 \). So, we set the quadratic function equal to zero and solve for \( x \):

\[
-0.2x^2 + 2.1x + 8 = 0
\]

I will solve this quadratic equation next.The two solutions for the quadratic equation are approximately \( x = -2.97 \) and \( x = 13.47 \). Since a negative distance doesn't make sense in this context, the valid solution is \( x = 13.47 \) feet.

Thus, the ball travels approximately **13.5 feet** horizontally before hitting the ground.

Would you like more details or have any questions?

Here are 5 related questions to further explore:
1. How is the vertex form of a quadratic function useful for finding the maximum height?
2. What would happen to the ball's trajectory if the initial height was increased?
3. How does the horizontal distance at maximum height relate to the total distance traveled?
4. What impact does the coefficient of \( x^2 \) have on the shape of the parabola?
5. How can we calculate the time it takes for the ball to reach the ground?

**Tip:** When solving quadratics, always discard the solution that doesn't fit the physical context (like negative distances or times).

A ball is thrown upward and outward from a height of 8 feet. The height of the ball, f(x), in feet, can be modeled by f(x) = -0.2x^2 + 2.1x + 8, where x is the ball's horizontal distance, in feet, from where it was thrown. Find the maximum height and the horizontal distance the ball travels before hitting the ground.

This problem models the height of a ball thrown upward using the quadratic function f(x) = -0.2x^2 + 2.1x + 8. Learn how to calculate the maximum height and total horizontal distance before the ball hits the ground. Ideal for students in grades 9-10.

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