1. **State the problem:** We have a parabolic curve representing the bridge arch given by the function $f(x) = px^2 + \frac{1}{4}x + 91$. The length of the base $AB$ is 144 m, and we want to find the height of the highest point of the curve above the road surface. 2. **Understand the problem:** The highest point of a parabola $f(x) = ax^2 + bx + c$ (with $a < 0$) is its vertex. The $x$-coordinate of the vertex is given by the formula: $$x = -\frac{b}{2a}$$ 3. **Identify known values:** Here, $a = p$, $b = \frac{1}{4}$, and $c = 91$. The points $A$ and $B$ lie on the road surface, so $f(x)$ at $A$ and $B$ equals the road surface height, which we can take as 0 for reference. The length $AB = 144$ m corresponds to the horizontal distance between $A$ and $B$. 4. **Set up the boundary conditions:** Let $A$ be at $x = 0$ and $B$ at $x = 144$. Since $A$ and $B$ are on the road surface, $$f(0) = p \cdot 0^2 + \frac{1}{4} \cdot 0 + 91 = 91$$ This means the curve is 91 m above the road at $x=0$, which contradicts the assumption that $A$ and $B$ are on the road surface. To fix this, we shift the coordinate system so that the road surface is at height 0, meaning the parabola is actually: $$f(x) = px^2 + \frac{1}{4}x + 91 - 91 = px^2 + \frac{1}{4}x$$ 5. **Apply the condition that $f(0) = 0$ and $f(144) = 0$:** $$f(0) = 0 \Rightarrow 0 = p \cdot 0^2 + \frac{1}{4} \cdot 0 = 0$$ $$f(144) = 0 \Rightarrow 0 = p \cdot 144^2 + \frac{1}{4} \cdot 144$$ 6. **Solve for $p$:** $$0 = 20736p + 36$$ $$20736p = -36$$ $$p = -\frac{36}{20736} = -\frac{1}{576}$$ 7. **Find the vertex $x$-coordinate:** $$x = -\frac{b}{2a} = -\frac{\frac{1}{4}}{2 \cdot (-\frac{1}{576})} = -\frac{\frac{1}{4}}{-\frac{1}{288}} = \frac{1}{4} \times 288 = 72$$ 8. **Calculate the height at the vertex:** $$f(72) = -\frac{1}{576} \cdot 72^2 + \frac{1}{4} \cdot 72$$ Calculate $72^2 = 5184$: $$f(72) = -\frac{5184}{576} + 18 = -9 + 18 = 9$$ 9. **Add back the 91 m offset:** The original function had a +91, so the actual height above the road surface is: $$9 + 91 = 100$$ **Final answer:** The highest point of the curve is 100 meters above the road surface.