Geometric Optimization Math Example 3

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Example 3

easy

Two rectangles have the same perimeter of

24

cm: one is

8 \times 4

cm and one is

6 \times 6

cm. Which has greater area? Does this match the

P^2/16

maximum?

Solution

1
Step 1: Area of $8\times4 = 32$ cm $^2$ . Area of $6\times6 = 36$ cm $^2$ . The square has greater area.
2
Step 2: $P^2/16 = 576/16 = 36$ cm $^2$ . The square ( $6\times6$ ) achieves the maximum. ✓

Answer

The

6\times6

square has greater area (

36

^2

32

^2

); it equals the maximum

P^2/16 = 36

^2

The square maximises area for a fixed perimeter. Any non-square rectangle with the same perimeter has strictly less area, confirming that among all rectangles with equal perimeter, the square is optimal.

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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Example 4 hard

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Area Perimeter