1. **State the problem:** We are given the height of a water stream from a fire hose as a function of horizontal distance $d$: $$h = -0.5d(d - 24).$$ We need to find the horizontal distance between the two points where the height is 54 meters. 2. **Rewrite the height equation:** $$h = -0.5d^2 + 12d$$ 3. **Set height to 54 meters and solve for $d$: ** $$54 = -0.5d^2 + 12d$$ 4. **Rearrange the equation:** $$-0.5d^2 + 12d - 54 = 0$$ 5. **Multiply entire equation by $-2$ to clear decimals:** $$d^2 - 24d + 108 = 0$$ 6. **Use the quadratic formula:** $$d = \frac{24 \pm \sqrt{(-24)^2 - 4 \times 1 \times 108}}{2}$$ 7. **Calculate the discriminant:** $$\sqrt{576 - 432} = \sqrt{144} = 12$$ 8. **Find the two solutions:** $$d_1 = \frac{24 - 12}{2} = \frac{12}{2} = 6$$ $$d_2 = \frac{24 + 12}{2} = \frac{36}{2} = 18$$ 9. **Calculate the horizontal distance between these points:** $$18 - 6 = 12$$ **Final answer:** The horizontal distance between the two points where the water is 54 meters high is 12 meters.