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COMEDK2024MathematicsApplication of DerivativesActual

The dimensions of the largest rectangle of side x and y that can be inscribed in the right angled triangle of sides a and b is

Options

  1. Aa 4 , b 4
  2. Ba, b
  3. Ca 2 , b 2
  4. D3 a 2 , 3 b 2

Correct answer

C. a 2 , b 2

Step-by-step solution

Let the right-angled triangle have base a and height b . The rectangle is inscribed such that one vertex lies on the hypotenuse, and two sides lie along the legs of the triangle. Let the rectangle have width x and height y . Using similar triangles, the small triangle at the top has base x and height b - y . The large triangle has base a and height b . By similarity: x a = b - y b This simplifies to x = a(b - y) b = a - a b y . The area A of the rectangle is given by A = xy = y(a - a b y) = ay - a b y^2 . To maximi

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