Question

A ball was kicked from the ground with a velocity of 64 feet per second and followed the path modeled by the function: h (t) = -16t^2 + 64t
The vertex of the rock's path indicates the rock took BLANK seconds to reach its maximum height of BLANK feet. Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball BLANK.
Options for each BLANK:
hits the ground after 4 seconds, 2, reaches a maximum height at 4 feet, 4, 64, 0, 48

Answer 1

To find the time it takes for the ball to reach its maximum height, we can determine the vertex of the parabola modeled by the function \( h(t) = -16t^2 + 64t \).

The vertex of a parabola in standard form \( h(t) = at^2 + bt + c \) can be found using the formula \( t = -\frac{b}{2a} \).

In this case:
- \( a = -16 \)
- \( b = 64 \)

Calculating \( t \):
\[
t = -\frac{64}{2 \cdot -16} = -\frac{64}{-32} = 2 \text{ seconds}
\]

Substituting \( t = 2 \) back into the equation to find the maximum height:
\[
h(2) = -16(2)^2 + 64(2) = -16(4) + 128 = -64 + 128 = 64 \text{ feet}
\]

Now, for the options:
1. "The vertex of the rock's path indicates the rock took **2** seconds to reach its maximum height of **64** feet."
2. "Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball **hits the ground after 4 seconds.**"

So, filling in the blanks:
The vertex of the rock's path indicates the rock took **2** seconds to reach its maximum height of **64** feet. Looking at the graph of this function, we can see the domain in context to this situation is 0 to 4 because the ball **hits the ground after 4 seconds.**