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CAT 2023 Slot 3 QA Question 18

Multiple choice (+3 / −1) · Geometry · Circles · Try it, then check the answer and solution below.

CAT 2023 Slot 3QAGeometry • CirclesModerate
A rectangle with the largest possible area is drawn inside a semicircle of radius 2 cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is
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Answer and solution

Answer: B) 2:12:1

📌 Core Concept
To maximize the area of a rectangle inscribed in a semicircle of radius $r , we use the relationship between the rectangle's dimensions and the semicircle's equation. The area is maximized when the rectangle's width and height satisfy the semicircle's equation.
🔢 Step-by-Step Solution
1
Define Variables:
Let the semicircle have radius r=2r = 2 \text{ cm}.
The rectangle has width 2x2x and height yy.
2
Equation of the Semicircle:
The equation is x2+y2=r2x^2 + y^2 = r^2. Substituting r=2r = 2: x2+y2=4x^2 + y^2 = 4 Solving for yy: y=4−x2y = \sqrt{4 - x^2}$
3
Area of the Rectangle:
A=2x⋅y=2x⋅4−x2A = 2x \cdot y = 2x \cdot \sqrt{4 - x^2}
4
Maximize the Area:
Take the derivative of AA with respect to xx:
dAdx=24−x2−2x24−x2\frac{dA}{dx} = 2\sqrt{4 - x^2} - \frac{2x^2}{\sqrt{4 - x^2}}
Set the derivative to zero and solve for xx:
\[ 2

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