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: Parametrize the rectangle by angle.
Let the rectangle be inscribed in the circle of radius \(R=10\) with its diagonal along the diameter. If \(\theta\) is the angle a diagonal half makes with a side, the sides can be written as \(2R\cos\theta\) and \(2R\sin\theta\).

Step 2: Write the area in terms of \(\theta\).
\[ A(\theta) = (2R\cos\theta)(2R\sin\theta) = 2R^2\sin2\theta \]

Step 3: Maximize using the sine bound.
Since \(\sin2\theta\leq1\), the area is largest when \(\sin2\theta=1\), that is at \(\theta=\pi/4\), which makes the rectangle a square.

Step 4: Compute the maximum value.
\[ A_{max} = 2R^2 = 2(10)^2 = 200 \]

Step 5: Final conclusion.
Maximum area is \[ \boxed{200 \text{ cm}^2} \]