Question

A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equation: h = -16² + 60t + 0, where h is the height in feet, and t is the time in seconds. Using this equation, what is the maximum range the ball will reach? Round to the nearest hundredth.

Answer 1

To find the maximum height reached by the football after it is kicked, we can derive it from the given quadratic equation for height:

\[
h(t) = -16t^2 + 60t + 0
\]

This is a quadratic equation in the standard form \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 60 \)
- \( c = 0 \)

The maximum value of a quadratic function occurs at the vertex, which can be found using the formula:

\[
t = -\frac{b}{2a}
\]

Substituting the values for \( a \) and \( b \):

\[
t = -\frac{60}{2(-16)} = -\frac{60}{-32} = \frac{60}{32} = 1.875 \text{ seconds}
\]

Next, we can substitute this value of \( t \) back into the height equation to find the maximum height:

\[
h(1.875) = -16(1.875)^2 + 60(1.875)
\]

Calculating \( (1.875)^2 \):

\[
(1.875)^2 = 3.515625
\]

Now substitute this back into the equation for height:

\[
h(1.875) = -16(3.515625) + 60(1.875)
\]
\[
h(1.875) = -56.25 + 112.5
\]
\[
h(1.875) = 56.25 \text{ feet}
\]

Thus, the maximum height that the ball will reach is **56.25 feet**.