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?

(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)
left parenthesis 51.375 comma 2.75 right parenthesis

(5.676,0)

Answer 1

To find the vertex of the quadratic function \( h = -6t^2 + 33t + 6 \), we can use the vertex formula for a parabola given by the equation \( ax^2 + bx + c \). The x-coordinate (in this case, time \( t \)) of the vertex can be found using the formula:

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

In this equation, \( a = -6 \) and \( b = 33 \).

Calculating:

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

Now, we will substitute \( t = 2.75 \) back into the original height function to find the corresponding height \( h \):

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

Calculating \( (2.75)^2 \):

\[
(2.75)^2 = 7.5625
\]

Substituting back into the height formula:

\[
h = -6(7.5625) + 33(2.75) + 6
\]

Calculating each term:

\[
-6(7.5625) = -45.375
\]
\[
33(2.75) = 90.75
\]

Now, adding these values:

\[
h = -45.375 + 90.75 + 6
\]
\[
h = -45.375 + 96.75
\]
\[
h = 51.375
\]

Thus, the vertex of the function (the maximum point of the parabola) is at:

\[
(2.75, 51.375)
\]

So the correct ordered pair representing the vertex is:

\((2.75, 51.375)\)