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 = −16t^2 + 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.
____ feet

Answer 1

To find the maximum height (or the range) of the ball using the equation \( h = -16t^2 + 60t + 0 \), we can use the vertex formula for a quadratic equation, which gives the time at which the maximum height occurs. For a quadratic equation in the form \( h = at^2 + bt + c \), the \( t \) coordinate of the vertex is given by:

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

In this case, \( a = -16 \) and \( b = 60 \).

Plugging in these values:

\[
t = -\frac{60}{2 \times -16} = -\frac{60}{-32} = \frac{60}{32} = \frac{15}{8} = 1.875 \text{ seconds}
\]

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

\[
h = -16(1.875)^2 + 60(1.875) + 0
\]

Calculating \( (1.875)^2 \):

\[
(1.875)^2 = 3.515625
\]

Now substituting back into the equation:

\[
h = -16(3.515625) + 60(1.875)
\]

Calculating each part:

\[
-16(3.515625) = -56.25
\]
\[
60(1.875) = 112.5
\]

Now adding these results:

\[
h = -56.25 + 112.5 = 56.25 \text{ feet}
\]

Thus, the maximum height (or range) the ball will reach is:

\[
\boxed{56.25} \text{ feet}
\]