1. **Problem Statement:** We are given the equation for the height of an arch above water as a function of horizontal distance $x$ from the river's center: $$h = -0.045x^2 + 12.5$$ We want to analyze this quadratic function to understand the height of the arch at different points. 2. **Formula and Important Rules:** This is a quadratic function of the form: $$h = ax^2 + bx + c$$ where $a = -0.045$, $b = 0$, and $c = 12.5$. Key points about quadratic functions: - The graph is a parabola. - Since $a < 0$, the parabola opens downward. - The vertex represents the maximum point. - The vertex $x$-coordinate is given by: $$x = -\frac{b}{2a}$$ 3. **Find the Vertex:** Since $b=0$, $$x = -\frac{0}{2(-0.045)} = 0$$ 4. **Calculate the Maximum Height:** Substitute $x=0$ into the equation: $$h = -0.045(0)^2 + 12.5 = 12.5$$ So, the maximum height of the arch is 12.5 meters at the center of the river. 5. **Find the Intercepts (where height is zero):** Set $h=0$: $$0 = -0.045x^2 + 12.5$$ Rearranged: $$0.045x^2 = 12.5$$ $$x^2 = \frac{12.5}{0.045} \approx 277.78$$ $$x = \pm \sqrt{277.78} \approx \pm 16.67$$ This means the arch touches the water at approximately 16.67 meters to the left and right of the center. 6. **Summary:** - The arch is highest at the center ($x=0$) with height 12.5 meters. - The arch meets the water at about $x = \pm 16.67$ meters. - The parabola is symmetric about the center. Final answer: The maximum height of the arch is **12.5 meters** at the center, and the arch touches the water at approximately **16.67 meters** from the center on both sides.