Question

Maximum area of the rectangle inscribed in a circle of radius 10 cm is

• A. 100
• B. 200
• C. 250
• D. 150

Hint:

For rectangle inscribed in circle, use derivative method with constraint \(x^2 + y^2 = R^2\) to maximize area.

Answer 1

The Correct Option is B

Step 1: Area of rectangle.
Let rectangle sides \(2x, 2y\). Area \(A = 2x \cdot 2y = 4xy\)

Step 2: Circle constraint.
Circle radius \(R = 10\), so rectangle inside circle: \(x^2 + y^2 = R^2 = 100\)

Step 3: Express area in terms of single variable.
\(y = \sqrt{R^2 - x^2}\)
\(A(x) = 4x \sqrt{R^2 - x^2}\)

Step 4: Maximize area using derivative.
\(\frac{dA}{dx} = 4\sqrt{R^2 - x^2} + 4x \cdot \frac{-x}{\sqrt{R^2 - x^2}} = \frac{4(R^2 - 2x^2)}{\sqrt{R^2 - x^2}} = 0 \implies x^2 = R^2/2 = 50\)

Step 5: Compute maximum area.
\(x = y = \sqrt{50} \implies A_{max} = 4xy = 4 \cdot 50 = 200\)

Step 6: Final conclusion.
Hence, maximum area is \[ \boxed{200 \text{ cm}^2} \]