1. **State the problem:** We have a parabolic arch bridge 50 ft high at the center and 200 ft wide at the base. We want to find the height of the arch 30 ft from the center to see if a 30 ft tall boat can pass underneath. 2. **Set up the coordinate system and equation:** Place the origin at the center of the arch, so the vertex of the parabola is at $(0,50)$. The parabola opens downward because the arch height decreases away from the center. 3. **Use the vertex form of a parabola:** $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex. Here, $h=0$, $k=50$, so: $$y = a x^2 + 50$$ 4. **Find $a$ using the base width:** The arch base is 200 ft wide, so the parabola crosses the water surface (height 0) at $x = \pm 100$: $$0 = a(100)^2 + 50$$ $$0 = 10000a + 50$$ $$a = -\frac{50}{10000} = -0.005$$ 5. **Write the full equation:** $$y = -0.005 x^2 + 50$$ 6. **Find the height of the arch 30 ft from the center:** $$y = -0.005 (30)^2 + 50 = -0.005 \times 900 + 50 = -4.5 + 50 = 45.5$$ 7. **Compare the arch height to the boat height:** The arch height at 30 ft from the center is 45.5 ft, which is greater than the boat height of 30 ft. **Conclusion:** The boat will clear the arch when it is 30 ft from the center. Final answer: The boat clears the arch with about 15.5 ft of clearance.