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. What is the number of feet in the maximum height of the football?

Answer

To find the maximum height of the football modeled by the function \( h(x) = -\frac{1}{225}x^2 + \frac{2}{3}x \), we can follow these steps:

### Step 1: Identify the function
The function given is:
\[ h(x) = -\frac{1}{225}x^2 + \frac{2}{3}x \]
This is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -\frac{1}{225} \) and \( b = \frac{2}{3} \).

### Step 2: Find the derivative
To find the maximum height, we first need to find the derivative of \( h(x) \):
\[ h'(x) = \frac{d}{dx} \left(-\frac{1}{225}x^2 + \frac{2}{3}x\right) \]
Using the power rule, we differentiate:
\[ h'(x) = -\frac{2}{225}x + \frac{2}{3} \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ -\frac{2}{225}x + \frac{2}{3} = 0 \]
Rearranging gives:
\[ \frac{2}{3} = \frac{2}{225}x \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):
\[ x = \frac{225}{3} = 75 \]

### Step 5: Substitute \( x \) back into \( h(x) \)
Now that we have the value of \( x \), we substitute it back into the original function to find the maximum height:
\[ h(75) = -\frac{1}{225}(75^2) + \frac{2}{3}(75) \]
Calculating \( 75^2 \):
\[ 75^2 = 5625 \]
Now substituting:
\[ h(75) = -\frac{1}{225}(5625) + \frac{2}{3}(75) \]
Calculating the first term:
\[ -\frac{5625}{225} = -25 \]
Calculating the second term:
\[ \frac{2}{3}(75) = 50 \]
Now, combining these results:
\[ h(75) = -25 + 50 = 25 \]

### Conclusion
The maximum height of the football is:
\[ \boxed{25} \text{ feet} \]