Question refer to the equation below.
h(x)=-(1)/(225)x^(2)+(2)/(3)x`
The function h above models the path of a football when it is kicked during an attempt to make a field goal where x is the horizontal distance, in feet, from the kick, and h(x) is the corresponding height of the football, in feet above the ground.
Q. After the ball is kicked, what is the number of feet the football travels horizontally before it hits the ground?

Answer

To find the horizontal distance the football travels before it hits the ground, we need to determine when the height \( h(x) \) is equal to zero. This means we will set the equation \( h(x) \) to zero and solve for \( x \).

1. **Set the height function to zero:**
\[ h(x) = -\frac{1}{225}x^2 + \frac{2}{3}x = 0 \]

2. **Multiply through by 225 to eliminate the fraction:**
\[ 225 \left(-\frac{1}{225}x^2 + \frac{2}{3}x\right) = 0 \]
This simplifies to:
\[ -x^2 + 150x = 0 \]

3. **Factor out \( x \):**
\[ x(-x + 150) = 0 \]

4. **Set each factor to zero:**
- From \( x = 0 \), we find one solution.
- From \( -x + 150 = 0 \), we solve for \( x \):
\[ -x + 150 = 0 \implies x = 150 \]

5. **Interpret the solutions:**
- \( x = 0 \) represents the point where the football is kicked.
- \( x = 150 \) represents the point where the football hits the ground.

Thus, the football travels **150 feet** horizontally before it hits the ground.