https://file.aiquickdraw.com/mathbot/file/1726680887680ybb11agc.jpg

The function given for the soccer ball's arc is \( h(x) = -8x^2 + 64x \), where \( h(x) \) represents the height of the ball, and \( x \) is the horizontal distance. To find out when the ball hits the ground, we need to determine the value of \( x \) when \( h(x) = 0 \), since the ball is on the ground when its height is zero.

We can solve for \( x \) by setting the equation equal to zero:

\[
-8x^2 + 64x = 0
\]

Factor the quadratic equation:

\[
-8x(x - 8) = 0
\]

This gives two solutions:

\[
x = 0 \quad \text{or} \quad x = 8
\]

So, the ball starts at \( x = 0 \) (the initial point) and hits the ground again at \( x = 8 \).

Therefore, the ball hits the ground at the point \( (8, 0) \).

### The correct answer is: \( (8, 0) \).

Would you like more details or have any further questions?

Here are 5 related questions to explore:
1. What is the maximum height the ball reaches, and how can you find it?
2. How would the graph of the function \( h(x) = -8x^2 + 64x \) look like?
3. How do you find the vertex of a quadratic function like this one?
4. What would happen if the coefficient of \( x^2 \) were positive instead of negative?
5. How does changing the values of the coefficients affect the trajectory of the ball?

**Tip:** The point where the ball reaches its maximum height can be found using the vertex formula for a quadratic equation.

A soccer ball is kicked from the ground in an arc defined by the function, h(x) = -8x^2 + 64x. At what point does the ball hit the ground?

Find out how to solve a quadratic equation that represents the arc of a soccer ball. The equation h(x) = -8x^2 + 64x describes the ball's height, and we calculate when it hits the ground. Suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation