Geometric Optimization Math Example 2

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Example 2

easy
A rectangle has perimeter P=40P = 40 cm. Using the formula maximum area =P2/16= P^2/16, compute the maximum area and the dimensions of the optimal rectangle.

Solution

  1. 1
    Step 1: Maximum area =P216=40216=160016=100= \dfrac{P^2}{16} = \dfrac{40^2}{16} = \dfrac{1600}{16} = 100 cm2^2.
  2. 2
    Step 2: The maximum occurs for a square. Each side =P/4=10= P/4 = 10 cm.
  3. 3
    Step 3: Verify: 10×10=10010 \times 10 = 100 cm2^2. ✓

Answer

Maximum area =100= 100 cm2^2; optimal shape is a 10 cm×10 cm10\,\text{cm}\times10\,\text{cm} square.
For a fixed perimeter, the rectangle with maximum area is always a square. The formula Amax⁡=P2/16A_{\max} = P^2/16 encodes this fact. This result is a consequence of the AM-GM inequality: area is maximised when length equals width.

About Geometric Optimization

Finding the best geometric configuration — the shape that maximizes area, minimizes perimeter, uses the least material, or achieves some other optimal outcome — subject to given constraints.

Learn more about Geometric Optimization →

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