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A football player kicks the ball with an initial upward velocity of 60 feet per second. This situation is modeled using the equ...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.(1 point)
feet
, 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.(1 point)
feet
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Answered by
GPT-4o mini
To determine the maximum height the football will reach, we can use the vertex formula for a quadratic equation in the standard form \( h = at^2 + bt + c \). In this case, \( a = -16 \), \( b = 60 \), and \( c = 0 \).
The formula to find the time \( t \) at which the maximum height occurs is given by:
\[
t = -\frac{b}{2a}
\]
Plugging in the values of \( a \) and \( b \):
\[
t = -\frac{60}{2 \times -16} = \frac{60}{32} = 1.875 \text{ seconds}
\]
Next, we substitute \( t = 1.875 \) back into the height equation \( h = -16t^2 + 60t + 0 \) to find the maximum height:
\[
h = -16(1.875)^2 + 60(1.875)
\]
Calculating \( (1.875)^2 \):
\[
(1.875)^2 = 3.515625
\]
Now substitute this value into the height equation:
\[
h = -16(3.515625) + 60(1.875)
\]
Calculating \( -16(3.515625) \):
\[
-16(3.515625) = -56.25
\]
Calculating \( 60(1.875) \):
\[
60(1.875) = 112.5
\]
Now combine these results:
\[
h = -56.25 + 112.5 = 56.25 \text{ feet}
\]
Thus, the maximum height that the ball will reach is:
\[
\boxed{56.25} \text{ feet}
\]
The formula to find the time \( t \) at which the maximum height occurs is given by:
\[
t = -\frac{b}{2a}
\]
Plugging in the values of \( a \) and \( b \):
\[
t = -\frac{60}{2 \times -16} = \frac{60}{32} = 1.875 \text{ seconds}
\]
Next, we substitute \( t = 1.875 \) back into the height equation \( h = -16t^2 + 60t + 0 \) to find the maximum height:
\[
h = -16(1.875)^2 + 60(1.875)
\]
Calculating \( (1.875)^2 \):
\[
(1.875)^2 = 3.515625
\]
Now substitute this value into the height equation:
\[
h = -16(3.515625) + 60(1.875)
\]
Calculating \( -16(3.515625) \):
\[
-16(3.515625) = -56.25
\]
Calculating \( 60(1.875) \):
\[
60(1.875) = 112.5
\]
Now combine these results:
\[
h = -56.25 + 112.5 = 56.25 \text{ feet}
\]
Thus, the maximum height that the ball will reach is:
\[
\boxed{56.25} \text{ feet}
\]
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