1. **State the problem:** We have a curve representing the arch of the Zhiropisny bridge given by the function $f(x) = px^2 + \frac{1}{4}x + 91$. The length of the base $AB$ is 144 meters. We need to find the height of the highest point of the curve from the road surface. 2. **Understand the problem:** The highest point on a parabola $f(x) = ax^2 + bx + c$ (where $a < 0$ for a maximum) is at the vertex. The $x$-coordinate of the vertex is given by the formula: $$x = -\frac{b}{2a}$$ 3. **Find $p$ using the length of $AB$:** Since $AB$ is the base of the arch, assume $A$ and $B$ are the roots of $f(x)$ where the curve meets the road surface (height = 0). So, $$f(x) = 0 = px^2 + \frac{1}{4}x + 91$$ The distance between roots $A$ and $B$ is 144 meters, so: $$|x_2 - x_1| = 144$$ 4. **Calculate the roots:** The roots of the quadratic are: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-\frac{1}{4} \pm \sqrt{\left(\frac{1}{4}\right)^2 - 4p \cdot 91}}{2p}$$ The distance between roots is: $$|x_2 - x_1| = \frac{\sqrt{b^2 - 4ac}}{|a|} = \frac{\sqrt{\left(\frac{1}{4}\right)^2 - 4p \cdot 91}}{|p|} = 144$$ 5. **Solve for $p$:** $$\sqrt{\frac{1}{16} - 364p} = 144|p|$$ Square both sides: $$\frac{1}{16} - 364p = 20736 p^2$$ Rearranged: $$20736 p^2 + 364 p - \frac{1}{16} = 0$$ Multiply entire equation by 16 to clear fraction: $$331776 p^2 + 5824 p - 1 = 0$$ 6. **Use quadratic formula for $p$:** $$p = \frac{-5824 \pm \sqrt{5824^2 - 4 \cdot 331776 \cdot (-1)}}{2 \cdot 331776}$$ Calculate discriminant: $$5824^2 = 33918776$$ $$4 \cdot 331776 = 1327104$$ $$\text{Discriminant} = 33918776 + 1327104 = 35245880$$ $$p = \frac{-5824 \pm \sqrt{35245880}}{663552}$$ $$\sqrt{35245880} \approx 5937.7$$ Two possible values: $$p_1 = \frac{-5824 + 5937.7}{663552} \approx \frac{113.7}{663552} \approx 0.000171$$ $$p_2 = \frac{-5824 - 5937.7}{663552} \approx \frac{-11761.7}{663552} \approx -0.0177$$ Since the arch is concave down (highest point), $p$ must be negative: $$p = -0.0177$$ 7. **Find the vertex $x$-coordinate:** $$x = -\frac{b}{2a} = -\frac{\frac{1}{4}}{2p} = -\frac{0.25}{2 \times -0.0177} = -\frac{0.25}{-0.0354} = 7.06$$ 8. **Calculate the height at vertex:** $$f(7.06) = p(7.06)^2 + \frac{1}{4} (7.06) + 91$$ $$= -0.0177 \times 49.84 + 1.765 + 91$$ $$= -0.882 + 1.765 + 91 = 91.883$$ **Answer:** The highest point of the curve is approximately $91.88$ meters above the road surface.