To find when the ball reaches its maximum height and what that maximum height is, we need to consider the quadratic function given for the height [tex]\( H(t) \)[/tex] of the ball:
[tex]\[ H(t) = -16t^2 + 32t + 5 \][/tex]
This is a quadratic equation of the form [tex]\( H(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 32 \)[/tex], and [tex]\( c = 5 \)[/tex]. Because the parabola opens downward (since [tex]\( a \)[/tex] is negative), the vertex represents the maximum point.
The time [tex]\( t \)[/tex] at which the ball reaches its maximum height can be found using the formula for the vertex of a parabola:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the given coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ t = -\frac{32}{2 \cdot -16} \][/tex]
[tex]\[ t = -\frac{32}{-32} \][/tex]
[tex]\[ t = 1 \][/tex]
So, the ball reaches its maximum height after [tex]\( 1 \)[/tex] second.
Next, to find the maximum height, we substitute [tex]\( t = 1 \)[/tex] back into the original height function [tex]\( H(t) \)[/tex]:
[tex]\[ H(1) = -16(1)^2 + 32(1) + 5 \][/tex]
[tex]\[ H(1) = -16(1) + 32(1) + 5 \][/tex]
[tex]\[ H(1) = -16 + 32 + 5 \][/tex]
[tex]\[ H(1) = 21 \][/tex]
Therefore, the maximum height is [tex]\( 21 \)[/tex] feet.
In conclusion,
after [tex]\( 1 \)[/tex] second, the ball will reach a maximum height of [tex]\( 21 \)[/tex] feet.
