Question

paulo kicks a soccer ball into the air with an initial upward velocity of 64 feet per second. the situation is modeled by the equation ( h = -16t^2 + 64t + 4 ) where ( h ) represents the height of the ball in feet and ( t ) represents the time in seconds. (2 points) use the vertex formula to complete the following statements. it takes (\boldsymbol{downarrow}) second(s) for the ball to reach the maximum height. the maximum height of the ball is (\boldsymbol{downarrow}) feet.

Explanation:

Step1: Recall vertex formula for parabola

For a quadratic function in the form \( h(t) = at^2 + bt + c \), the time \( t \) at the vertex (which gives the maximum or minimum value) is given by \( t = -\frac{b}{2a} \). Here, \( a = -16 \) and \( b = 64 \).
So, \( t = -\frac{64}{2\times(-16)} = -\frac{64}{-32} = 2 \) seconds.

Step2: Find maximum height

Substitute \( t = 2 \) into the height equation \( h(t) = -16t^2 + 64t + 4 \).
\( h(2) = -16\times(2)^2 + 64\times2 + 4 = -16\times4 + 128 + 4 = -64 + 128 + 4 = 68 \) feet.

Answer:

It takes 2 second(s) for the ball to reach the maximum height.
The maximum height of the ball is 68 feet.