1. **Problem statement:** The cable of a suspension bridge forms a parabola. The distance between towers is 150 m, the cable height at towers is 22 m, and the lowest point is 7 m above the roadway. Find the vertical distance from the roadway to the cable 15 m from the foot of a tower. 2. **Set up coordinate system:** Place the origin at the lowest point of the cable (vertex of the parabola). The towers are then at $x = \pm 75$ m (half of 150 m). 3. **Parabola equation:** The cable shape is a parabola with vertex at $(0,7)$, so the equation is $$y = a x^2 + 7$$ where $y$ is the height above the roadway. 4. **Use tower points to find $a$:** At $x = 75$, $y = 22$, so $$22 = a (75)^2 + 7$$ $$22 - 7 = 5625 a$$ $$15 = 5625 a$$ $$a = \frac{15}{5625} = \frac{1}{375}$$ 5. **Final parabola equation:** $$y = \frac{1}{375} x^2 + 7$$ 6. **Find height at $x=15$ m:** $$y = \frac{1}{375} (15)^2 + 7 = \frac{225}{375} + 7 = 0.6 + 7 = 7.6$$ 7. **Vertical distance from roadway:** The cable is 7.6 m above the roadway at 15 m from the tower. **Answer:** 7.6 m is not among the options, so check if the question asks for vertical distance to cable from roadway or from tower base. Since the cable height at towers is 22 m, the vertical distance from the tower base (22 m) minus cable height at 15 m is $$22 - 7.6 = 14.4$$ which is also not an option. Alternatively, the problem likely wants the height of the cable above the roadway at 15 m from the tower, which is 7.6 m. Since none of the options match 7.6 m, re-examine the problem: The lowest point is 7 m above roadway, so the cable height at 15 m from tower is 7.6 m above roadway. **Therefore, the vertical distance to the cable from the roadway 15 m from the tower is 7.6 m.** Since 7.6 m is not an option, the closest is 9.6 m (option B), but mathematically the answer is 7.6 m. **Slug:** cable_parabola **Subject:** physics