To find the vertex of the parabola represented by the function \( h = -6t^2 + 33t + 6 \), we can use the formula for the \( t \)-coordinate of the vertex, which is given by
\[ t = -\frac{b}{2a} \]
In this case, the coefficients are:
- \( a = -6 \)
- \( b = 33 \)
Plugging in the values, we find:
\[ t = -\frac{33}{2 \cdot -6} = -\frac{33}{-12} = \frac{33}{12} = 2.75 \]
Now, to find the \( h \)-coordinate of the vertex, we can substitute \( t = 2.75 \) back into the height function:
\[ h(2.75) = -6(2.75)^2 + 33(2.75) + 6 \]
Calculating \( (2.75)^2 \):
\[ (2.75)^2 = 7.5625 \]
Now substitute:
\[ h(2.75) = -6(7.5625) + 33(2.75) + 6 \] \[ = -45.375 + 90.75 + 6 \] \[ = -45.375 + 96.75 \] \[ = 51.375 \]
Thus, the vertex of the function is \((2.75, 51.375)\).
Therefore, the correct ordered pair representing the vertex is:
(2.75, 51.375).