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=−16t2+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 that the football reaches using the given equation \( h = -16t^2 + 60t + 0 \), we need to determine the vertex of the parabola represented by this quadratic equation.

The standard form of a quadratic equation is \( h = at^2 + bt + c \). For our equation:

- \( a = -16 \)
- \( b = 60 \)
- \( c = 0 \)

The time \( t \) at which the maximum height occurs is given by the formula:

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

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

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

Now we will substitute \( t = 1.875 \) back into the height equation to find the maximum height \( h \):

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

Calculating \( (1.875)^2 \):

\[
(1.875)^2 = 3.515625
\]

Now calculate \( h \):

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

Thus, the maximum height that the ball will reach is \( 56.25 \) feet.

So the final answer is:

\[
\boxed{56.25}
\]