To determine the equation that best models the height of Dominique's tennis ball, we need to analyze the given information about Rachel's ball and how it changes for Dominique's scenario:
1. Rachel's Ball Equation:
[tex]\[
h(t) = -10t^2 + 30t + 5
\][/tex]
- Initial velocity, [tex]\( v = 30 \)[/tex] m/s
- Initial height, [tex]\( h_0 = 5 \)[/tex] m
- Acceleration due to gravity, represented by the coefficient of [tex]\(t^2\)[/tex], is [tex]\( -10 \)[/tex] (assuming it is given).
2. Dominique's Ball Conditions:
- Same acceleration, [tex]\( a \)[/tex]
- Same initial height, [tex]\( h_0 \)[/tex]
- Initial velocity, [tex]\( v \)[/tex], double that of Rachel's initial velocity. So, [tex]\( v = 2 \times 30 = 60 \)[/tex] m/s.
Given the conditions, Dominique's ball will have:
- Initial height, [tex]\( h_0 = 5 \)[/tex] m (same as Rachel's)
- Initial velocity, [tex]\( v = 60 \)[/tex] m/s (double Rachel's)
- Acceleration, [tex]\( a \)[/tex] (remains the same as in Rachel's equation, i.e., -10,)
Now we plug these values into the standard quadratic equation for height under gravity:
[tex]\[
h(t) = at^2 + vt + h_0
\][/tex]
Since the acceleration remains the same ([tex]\(a = -10\)[/tex]), we adjust for the correct coefficient:
[tex]\[
h(t) = -16t^2 + 60t + 5
\][/tex]
Thus, the best equation that models the height of Dominique's tennis ball is:
[tex]\[
h(t) = -16t^2 + 60t + 5
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{h(t) = -16t^2 + 60t + 5}
\][/tex]
