Question

A volleyball is served from a height of 6 ft in an initial velocity of 33 ft per second the situation is modeled using the function h = -6tsquared plus 33t + 6 where h is the height and feet and t is the time and seconds using the function which ordered pair represents the vertex

Answer 1

To find the vertex of the quadratic function given by the equation \( h(t) = -6t^2 + 33t + 6 \), we can use the vertex formula for a parabola in the form \( ax^2 + bx + c \) where \( a = -6 \), \( b = 33 \), and \( c = 6 \).

The time \( t \) at which the vertex occurs can be calculated using the formula:

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

Substituting the values of \( a \) and \( b \):

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

Now that we have the value of \( t \), we can find the height \( h \) at this time by substituting \( t = 2.75 \) 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 substituting this back into the equation:

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

Thus, the vertex of the function, which represents the maximum height reached by the volleyball, is the ordered pair:

\[
(2.75, 51.375)
\]

So, the ordered pair representing the vertex is \( (2.75, 51.375) \).