To find the maximum height the ball reaches, we can use the vertex form of a quadratic function. The general form of the equation you provided is:
\[ h(t) = -16t^2 + 60t + 0 \]
This is a quadratic equation in the standard form \(h(t) = at^2 + bt + c\) where:
- \(a = -16\)
- \(b = 60\)
- \(c = 0\)
The maximum height occurs at the vertex of the parabola. For a quadratic function of the form \(h(t) = at^2 + bt + c\), the time \(t\) at which the maximum height occurs can be calculated using the formula:
\[ t = -\frac{b}{2a} \]
Substituting the values of \(a\) and \(b\):
\[ t = -\frac{60}{2 \times -16} = -\frac{60}{-32} = \frac{60}{32} = 1.875 \text{ seconds} \]
Next, we substitute \(t = 1.875\) back into the height equation to find the maximum height \(h(t)\):
\[ h(1.875) = -16(1.875)^2 + 60(1.875) + 0 \]
Calculating \( (1.875)^2 \):
\[ (1.875)^2 = 3.515625 \]
Now substituting this back into the equation:
\[ h(1.875) = -16(3.515625) + 60(1.875) \]
Calculating each term:
- \(-16 \times 3.515625 = -56.25\)
- \(60 \times 1.875 = 112.5\)
Now, combine them:
\[ h(1.875) = -56.25 + 112.5 = 56.25 \text{ feet} \]
Thus, the maximum height the ball will reach is:
\[ \boxed{56.25} \text{ feet} \]