A rectangle is inscribed in a semicircle of radius r with one side on the diameter. The maximum area of the rectangle is: A. r² B. r²/2 C. r²√2 D. 2r² Answer: A) r² Explanation: Let half-width = x, height = √(r² − x²). Area A = 2x√(r² − x²). dA/dx = 2√(r² − x²) − 2x²/√(r² − x²) = 0 → 2(r² − x²) − 2x² = 0 → r² = 2x² → x = r/√2. Max area = 2(r/√2) × r/√2 = r².

imo class 12 application of derivatives

A rectangle is inscribed in a semicircle of radius r with one side on the diameter. The maximum area of the rectangle is:

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A rectangle is inscribed in a semicircle of radius r with one side on the diameter. The maximum area of the rectangle is:

A. r²
B. r²/2
C. r²√2
D. 2r²

Answer: A) r²

Explanation: Let half-width = x, height = √(r² − x²). Area A = 2x√(r² − x²). dA/dx = 2√(r² − x²) − 2x²/√(r² − x²) = 0 → 2(r² − x²) − 2x² = 0 → r² = 2x² → x = r/√2. Max area = 2(r/√2) × r/√2 = r².

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