🐔 Chicken Coop Calculus 🐔

You have 100 feet of fencing to build a rectangular chicken coop against the side of your barn. What length should the side perpendicular to the barn (x) be to maximize the area (A)?

Play with different values to see how the area changes. Can you find the maximum without calculus?

How this relates to calculus:

This is an optimization problem! In calculus, we'd:

  1. Write an equation for the area in terms of x: A = x × y = x(100 - 2x)
  2. Take the derivative: dA/dx = 100 - 4x
  3. Find critical points by setting derivative to 0: 100 - 4x = 0 → x = 25
  4. Verify it's a maximum (2nd derivative test: d²A/dx² = -4 → concave down)

The maximum area occurs when x = 25 ft, y = 50 ft, giving A = 1250 ft².