To find the maximum height that the football reaches, we can analyze the given quadratic equation for height, which is:
\[ h(t) = -16t^2 + 60t + 0 \]
This equation is in the standard form \(h(t) = at^2 + bt + c\), where:
- \(a = -16\)
- \(b = 60\)
- \(c = 0\)
The maximum height of a quadratic function occurs at the vertex. The time \(t\) at which this maximum occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
Substituting the values of \(a\) and \(b\):
\[ t = -\frac{60}{2 \times -16} = -\frac{60}{-32} = \frac{60}{32} = \frac{15}{8} \text{ seconds} \]
Now we can find the maximum height by substituting this \(t\) value back into the height equation:
\[ h\left(\frac{15}{8}\right) = -16\left(\frac{15}{8}\right)^2 + 60\left(\frac{15}{8}\right) + 0 \]
Calculating \(\left(\frac{15}{8}\right)^2\):
\[ \left(\frac{15}{8}\right)^2 = \frac{225}{64} \]
Now substituting back into the height equation:
\[ h\left(\frac{15}{8}\right) = -16\left(\frac{225}{64}\right) + 60\left(\frac{15}{8}\right) \]
Calculating each term:
- \(-16\left(\frac{225}{64}\right) = -\frac{3600}{64} = -56.25\)
- \(60\left(\frac{15}{8}\right) = \frac{900}{8} = 112.5\)
Now add these two results together:
\[ h\left(\frac{15}{8}\right) = -56.25 + 112.5 = 56.25 \text{ feet} \]
Thus, the maximum height that the football will reach is 56.25 feet.
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