Question

A volleyball is served from a height of 6 feet with an initial velocity of 33 feet per second. The situation is modeled using the function h = -6t^2 + 33t + 6 where h is the height in feet and t is the time in seconds. Which ordered pairs represents the vertex
(0,6)
(51.375, 2.75)
(2.75, 51.375)
(6.676, 0)

Answer 1

To find the vertex of the quadratic function \( h(t) = -6t^2 + 33t + 6 \), we can use the vertex formula for a parabola given by \( t = -\frac{b}{2a} \), where \( h(t) = at^2 + bt + c \).

In this case:
- \( a = -6 \)
- \( b = 33 \)
- \( c = 6 \)

Now, we calculate the \( t \)-coordinate of the vertex:

\[
t = -\frac{b}{2a} = -\frac{33}{2 \times -6} = \frac{33}{12} = 2.75
\]

Next, we need to find the \( h \)-coordinate (height) at this time \( t = 2.75 \):

\[
h(2.75) = -6(2.75)^2 + 33(2.75) + 6
\]

Calculating \( (2.75)^2 \):

\[
(2.75)^2 = 7.5625
\]
\[
h(2.75) = -6(7.5625) + 33(2.75) + 6
\]
\[
= -45.375 + 90.75 + 6
\]
\[
= 51.375 - 45.375 = 6.375
\]

Thus, the vertex is at the point \( (2.75, 51.375) \).

Looking at the options provided:
- (0,6)
- (51.375, 2.75)
- (2.75, 51.375)
- (6.676, 0)

The correct ordered pair representing the vertex is:

\[
(2.75, 51.375)
\]