1. **Problem Statement:** We need to model the elevation $f(x)$ of a pedestrian bridge spanning 60 meters horizontally, peaking at 8 meters at the midpoint $x=30$, with slope constraints and smooth transitions. 2. **Constructing the function:** A smooth, symmetric function that starts and ends at 0 height and peaks at 8 can be modeled by a cosine-based function: $$f(x) = 8 \left(1 - \cos\left(\frac{\pi x}{60}\right)\right)/2$$ This function satisfies: - $f(0) = 0$ - $f(60) = 0$ - $f(30) = 8$ - Symmetry about $x=30$ 3. **Slope and incline check:** The slope is the derivative: $$f'(x) = 8 \cdot \frac{1}{2} \cdot \sin\left(\frac{\pi x}{60}\right) \cdot \frac{\pi}{60} = \frac{4\pi}{60} \sin\left(\frac{\pi x}{60}\right) = \frac{\pi}{15} \sin\left(\frac{\pi x}{60}\right)$$ The maximum slope magnitude is at $x=15$ or $x=45$: $$|f'(15)| = \frac{\pi}{15} \sin\left(\frac{\pi \cdot 15}{60}\right) = \frac{\pi}{15} \sin\left(\frac{\pi}{4}\right) = \frac{\pi}{15} \cdot \frac{\sqrt{2}}{2} \approx 0.148$$ The incline angle $\theta$ satisfies $\tan(\theta) = |f'(x)|$. Maximum incline angle: $$\theta_{max} = \arctan(0.148) \approx 8.45^\circ < 15^\circ$$ So the slope constraint is met. 4. **Limits:** - At midpoint $x=30$: $$\lim_{x \to 30} f(x) = f(30) = 8$$ - At ends $x=0$ and $x=60$: $$\lim_{x \to 0} f(x) = f(0) = 0$$ $$\lim_{x \to 60} f(x) = f(60) = 0$$ 5. **Continuity:** The function $f(x)$ is continuous on $[0,60]$ because it is composed of cosine functions which are continuous everywhere. Continuity ensures no abrupt jumps in elevation, which is critical for structural integrity and smooth transitions. 6. **Discontinuities or non-differentiability:** The function is differentiable everywhere on $(0,60)$ since cosine and sine are smooth functions. No discontinuities or sharp corners exist, so no unsafe slopes or stress points arise. **Final answer:** $$f(x) = 4 \left(1 - \cos\left(\frac{\pi x}{60}\right)\right)$$ with $$\lim_{x \to 30} f(x) = 8, \quad \lim_{x \to 0} f(x) = 0, \quad \lim_{x \to 60} f(x) = 0$$ The function is continuous and differentiable on $[0,60]$ with maximum slope less than $15^\circ$.