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The best shape (optimization)

Level 10 · Calculus · for questions like “You have 40 m of fence for a rectangle. What is the biggest area?”

💡 The idea

Write what you want to make biggest using one letter, take the derivative, and set it to 0.

✏️ Worked examples

1. 40 m of fence, a rectangle: if one side is x, the other is 20 − x. A = x(20 − x) = 20x − x². A′ = 20 − 2x = 0, so x = 10: a 10 by 10 square, 100 sq m.

2. Next to a river, 40 m for three sides: A = x(40 − 2x). A′ = 40 − 4x = 0, so x = 10. The field is 10 by 20, 200 sq m.

3. Two numbers add to 30. The biggest product? x(30 − x) is biggest at x = 15: 225.

⚠️ Watch out

🎯 Try these

  1. 60 m of fence, a rectangle: biggest area?
  2. River, 60 m for three sides: biggest area?
  3. Two numbers add to 50: biggest product?

Answers: 1) 225 2) 450 3) 625

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