To find the domain of the volleyball from the time it is served until it reaches its maximum height, we need to identify when the ball reaches its maximum height based on the given equation:
\[ h(t) = -16t^2 + 33t + 6 \]
The maximum height of a parabolic equation in the form \(h(t) = at^2 + bt + c\) occurs at the vertex, which is found using the formula:
\[ t = -\frac{b}{2a} \]
Here, \(a = -16\) and \(b = 33\). Plugging in these values:
\[ t = -\frac{33}{2 \cdot -16} = \frac{33}{32} \approx 1.03125 \text{ seconds} \]
Rounding this value, we can approximate the time it takes for the ball to reach maximum height to be around 1.03 seconds.
For the domain of the ball when it reaches its maximum height, it will be from \(t = 0\) (the time it is served) to \(t \approx 1.03\) seconds (the time it reaches maximum height).
Thus, the correct domain fitted in terms of the choices given is:
- \(0 - 1.03\) seconds
This indicates the time range between when the ball is served and when it reaches its maximum height.
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