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:
\((2.75, 51.375)\)