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Answer: 9 cm by 9 cm (a square)

Let the length be \(x\) cm and breadth \(y\) cm. Perimeter \(2(x + y) = 36\), so \(y = 18 - x\).

Area \(A = xy = x(18 - x) = 18x - x^2\).

\(\dfrac{dA}{dx} = 18 - 2x\) and \(\dfrac{d^2A}{dx^2} = -2\).

For extreme values, \(\dfrac{dA}{dx} = 0 \Rightarrow x = 9\). Since \(\dfrac{d^2A}{dx^2} = -2 \lt 0\), \(A\) is maximum at \(x = 9\).

Then \(y = 18 - 9 = 9\). The area is maximum when the rectangle is 9 cm long and 9 cm broad, i.e. it is a square; maximum area \(= 81\) sq. cm.

Doubtly AI · 29 Oct 2026 AI-verified · Nov 2026 Report
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