Problem 5 · 1999 AMC 8
Medium
Geometry & Measurement
perimeter-area
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
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Answer: D — 400 square feet.
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Hint 1 of 2
The fence never changes length, so the perimeter is the thing that stays fixed — the area is free to grow. Find that fixed perimeter first.
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Hint 2 of 2
The shape that holds the most area for a given perimeter is the most "square" one. Compute the square's side from the fence, then its area.
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Approach: perimeter is fixed; the rounder shape holds more area
- The fence length is locked: 2(50 + 10) = 120 ft. Reshaping can't change that, so the square has side 120 ÷ 4 = 30 ft and area 30² = 900 sq ft.
- The old long-thin rectangle held 50 × 10 = 500 sq ft, so the gain is 900 − 500 = 400 square feet.
- The principle: for a fixed perimeter, the more equal the sides, the more area — a square always beats a stretched-out rectangle. That's why squaring up a 50×10 sliver buys so much extra room.
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