1. **State the problem:** We have a parabola defined by the height function $$h = -0.5d(d - 24)$$ where $h$ is the height in meters and $d$ is the horizontal distance in meters. 2. **Understand the parabola:** This is a quadratic function in standard form. Expanding it gives: $$h = -0.5(d^2 - 24d) = -0.5d^2 + 12d$$ 3. **Vertex and maximum height:** Since the coefficient of $d^2$ is negative, the parabola opens downward, so the vertex represents the maximum height. The vertex $d$-coordinate is given by: $$d = -\frac{b}{2a} = -\frac{12}{2 \times (-0.5)} = -\frac{12}{-1} = 12$$ 4. **Maximum height:** Substitute $d=12$ into $h$: $$h = -0.5 \times 12 \times (12 - 24) = -0.5 \times 12 \times (-12) = 72$$ This matches the given maximum height. 5. **Height at $d=5$:** Substitute $d=5$: $$h = -0.5 \times 5 \times (5 - 24) = -0.5 \times 5 \times (-19) = 47.5$$ 6. **Height at 54 meters and horizontal distance between points:** We want to find $d$ such that $h=54$: $$54 = -0.5 d (d - 24)$$ Multiply both sides by 2: $$108 = -d^2 + 24d$$ Rearranged: $$d^2 - 24d + 108 = 0$$ 7. **Solve quadratic:** Use quadratic formula: $$d = \frac{24 \pm \sqrt{24^2 - 4 \times 1 \times 108}}{2} = \frac{24 \pm \sqrt{576 - 432}}{2} = \frac{24 \pm \sqrt{144}}{2} = \frac{24 \pm 12}{2}$$ 8. **Roots:** $$d_1 = \frac{24 - 12}{2} = 6$$ $$d_2 = \frac{24 + 12}{2} = 18$$ 9. **Distance between points:** $$18 - 6 = 12$$ This matches the given horizontal distance between points at height 54. 10. **Domain:** The parabola is defined for $0 \leq d \leq 24$ as given. **Final answers:** - Maximum height is 72 meters at $d=12$. - Height is 47.5 meters at $d=5$. - Height is 54 meters at $d=6$ and $d=18$, with horizontal distance 12 meters between these points.