Wavy Roads
Functions & Equations
Switchback roads use more distance to buy a gentler climb — the trade-off an engineer has to balance.
Introduction
Roads that switch back and forth up a steep hill trade extra distance for a gentler gradient, one that vehicles and cyclists can actually climb. Model the road as a curve, differentiate it to find the gradient at any point, and compare the wavy route's length with the straight-line distance. Working out the exact extra length needs arc length, which sits just past both syllabi — a numerical approximation, adding up lots of short straight segments, gets you there without the formal integral.
Guiding Questions
- Find a road that zigzags or waves up a steep hill. What problem is the engineer solving?
- Model the road as a curve and calculate its gradient along the route. What is the maximum gradient a car or cyclist can manage?
- How much extra length does the wavy route add compared with the straight line?
- Design your own route up a real hill profile that keeps the gradient under a chosen limit using the least distance.