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

(5.676,0)
left parenthesis 5.676 comma 0 right parenthesis

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

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

(51.375,2.75)

Answer 1

To find the vertex of the quadratic function \( h(t) = -6t^2 + 33t + 6 \), we can use the formula for the vertex of a quadratic function in the form \( h(t) = at^2 + bt + c \), where the vertex \( t \) coordinate is given by:

\[
t = -\frac{b}{2a}
\]

Here, \( a = -6 \) and \( b = 33 \). Plugging these values into the formula:

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

Now, we need to find the height \( h \) at \( t = 2.75 \). Plugging \( t = 2.75 \) back into the function:

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

Calculating \( 2.75^2 \):

\[
2.75^2 = 7.5625
\]

Now substitute this back in:

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

Thus, the vertex of the parabola, which gives the maximum height of the volleyball, is \((2.75, 51.375)\).

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

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