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=−6t2+33t+6
, where h is the height in feet and t is time in seconds. Using the function, which ordered pair represents the vertex?(1 point)
Responses

(0,6)
left parenthesis 0 comma 6 right parenthesis

(51.375,2.75)
left parenthesis 51.375 comma 2.75 right parenthesis

(2.75,51.375)
left parenthesis 2.75 comma 51.375 right parenthesis

(5.676,0)

Answer 1

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

Here, we identify:
- \( a = -6 \)
- \( b = 33 \)

Now, we can calculate the value of \( t \):

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

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

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

\[
(2.75)^2 = 7.5625
\]

Now substitute this into the function:

\[
h(2.75) = -6(7.5625) + 33(2.75) + 6
\]
\[
= -45.375 + 90.75 + 6
\]
\[
= -45.375 + 96.75
\]
\[
= 51.375
\]

Thus, the vertex of the function is \( (2.75, 51.375) \).

So, the correct ordered pair that represents the vertex is:

\[
\boxed{(2.75, 51.375)}
\]