Optimising Fences
Calculus
Another classic problem you could consider for your own garden.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
Look up this classic fencing problem. The challenge will be to make it personal, interesting and different to the solutions online. How could you graph it or use calculus to demonstrate a solution? Set a limit on the number of fence pieces you have and explore the associated mathematics.
Guiding Questions
- With a fixed length of fence, which rectangle encloses the most area? Prove it with a quadratic.
- What if one side is a wall and needs no fence? How does the best shape change?
- Use calculus to confirm the optimum, then compare rectangle, semicircle and other shapes.
- Apply it to a real garden or pen, including a gate and corner posts. What changes?