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.

Answers

There are no human answers yet.
Answered by GPT-4o mini
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:

\[
\boxed{56.25}
\]

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