To solve this, we need to find the time [tex]\( t \)[/tex] when the ball reaches its maximum height and determine the maximum height. This involves working with the given quadratic function for height [tex]\( s(t) = -16t^2 + 28t + 5 \)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of the quadratic equation: The quadratic function is given in the form [tex]\( s(t) = at^2 + bt + c \)[/tex], where:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 28 \)[/tex]
- [tex]\( c = 5 \)[/tex]
2. Determine the vertex of the parabola: Since the quadratic equation represents a parabola opening downward (because [tex]\( a < 0 \)[/tex]), the maximum height occurs at the vertex of the parabola. The time [tex]\( t \)[/tex] at which the maximum height is reached is given by the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
3. Calculate the time [tex]\( t \)[/tex]:
[tex]\[
t = -\frac{28}{2 \times -16}
\][/tex]
[tex]\[
t = -\frac{28}{-32}
\][/tex]
[tex]\[
t = 0.875
\][/tex]
4. Find the maximum height: Plug [tex]\( t = 0.875 \)[/tex] back into the height function [tex]\( s(t) \)[/tex]:
[tex]\[
s(0.875) = -16(0.875)^2 + 28(0.875) + 5
\][/tex]
[tex]\[
s(0.875) = -16(0.875^2) + 28(0.875) + 5
\][/tex]
[tex]\[
s(0.875) = -16(0.765625) + 28(0.875) + 5
\][/tex]
[tex]\[
s(0.875) = -12.25 + 24.5 + 5
\][/tex]
[tex]\[
s(0.875) = 17.25
\][/tex]
Therefore, the ball reaches its maximum height of [tex]\( 17.25 \)[/tex] feet.
The time at which the ball reaches its maximum height is [tex]\( 0.875 \)[/tex] seconds after it has been thrown.
### Answer:
The ball reaches its maximum height of [tex]\( 17.25 \)[/tex] feet,
[tex]\[
\boxed{17.25}
\][/tex]
0.875 seconds after the ball is thrown.
[tex]\[
\boxed{0.875}
\][/tex]
