Road Construction on a Conical Mountain
Finding the optimal route for a 60-ton truck
Problem Definition
We need to build a road on a perfect cone with a 1-mile height and 2-mile base diameter. The road must allow a 60-ton truck to travel from one side to the opposite side, entering and exiting at horizontal points.
Cone Specifications
Height: 1 mile
Base Diameter: 2 miles
Base Radius: 1 mile
Slant Height: √2 ≈ 1.414 miles
Vehicle Specifications
Weight: 60 tons
Max Recommended Gradient: 6-7%
Typical max speed on incline: 25-35 mph
Engineering Challenges
Building a road on a conical surface presents several unique challenges:
- The steep natural slope of the cone (45° angle)
- Maintaining a safe gradient for heavy vehicles
- Structural stability of the road bed
- Erosion control on sloped surfaces
- Construction material transportation
For a 60-ton truck, the road must have a consistent gradient not exceeding 6-7% to ensure safe operation.
Optimal Solution: Spiral Road Design
The best route is a spiral road that gradually ascends the cone with a constant gradient, circling the cone multiple times to reach the summit before descending on the opposite side.
7%
Maximum Gradient
28.6 miles
Total Road Length
20+
Spiral Loops
This design minimizes the gradient to approximately 7%, making it safe for heavy trucks while maintaining a reasonable road length. The spiral design also offers structural advantages as it distributes the load evenly around the cone.
For construction, engineers would need to reinforce the road bed with proper drainage and stabilization techniques to prevent erosion and ensure long-term durability.
No comments:
Post a Comment