Question

Kayden is throwing a shot put at a track meet. He is 4.1 feet tall. The shot put leaves Kayden's hand at a height of 6 feet. It moves with an upward velocity of 48 feet per second. Using the function h=−16t^2+48t+6, which of the following correctly identifies the start and the vertex, plus also explains its meaning? Round to the nearest hundredth if necessary.(1 point)
Responses

A: (0,6); (1.5,42); The shot put leaves Kayden's hand at a distance of 0 feet and a height of 6
feet and will reach a maximum height of 42 feet 1.5 seconds after being thrown.
B: (6,0); (42,1.5) The shot put is thrown at a height of 6 feet. It will reach a maximum height of 42 feet 1.5 seconds after being thrown.
C: (4.1,0); (42,1.5) The shot put is thrown from a distance of 0 feet at a height of 4.1 feet and will land after 42 seconds.
D: (0,4.1) ; (1.5,42)

Answer 1

To analyze Kayden's shot put scenario using the provided quadratic function \( h(t) = -16t^2 + 48t + 6 \):

1. **Starting Point**:
- The initial height of the shot put when \( t = 0 \):
\[
h(0) = -16(0)^2 + 48(0) + 6 = 6
\]
Thus, the start point is \((0, 6)\).

2. **Vertex Calculation**:
- The vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by \( t = -\frac{b}{2a} \).
- Here, \( a = -16 \) and \( b = 48 \):
\[
t = -\frac{48}{2 \times -16} = \frac{48}{32} = 1.5
\]
- Now, substitute \( t = 1.5 \) back into the height function to find the maximum height:
\[
h(1.5) = -16(1.5)^2 + 48(1.5) + 6
\]
\[
h(1.5) = -16(2.25) + 72 + 6 = -36 + 72 + 6 = 42
\]
Thus, the vertex is \((1.5, 42)\).

### Summary of Results:
- The starting point is \((0, 6)\), which means the shot put leaves Kayden's hand at a height of 6 feet when time \( t = 0 \).
- The vertex point is \((1.5, 42)\), which indicates that the shot put will reach a maximum height of 42 feet at \( t = 1.5 \) seconds after being thrown.

### Correct Option:
Looking at the options:

**A: (0, 6); (1.5, 42); The shot put leaves Kayden's hand at a distance of 0 feet and a height of 6 feet and will reach a maximum height of 42 feet 1.5 seconds after being thrown.**

This option accurately describes both the starting point and the vertex, along with their meanings.

**Correct answer: A**