CAT 2023 Slot 3 QA Question & Solution
ArithmeticMedium
Question
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
Options
2 : 1
1 : 1
$\sqrt{5} : 1$
$\sqrt{2} : 1$
Solution
Let us assume the length of the rectangle is 'l' and breadth of the rectangle is 'b'.
The radius, l/2 and b in the above diagram form a right-angled triangle.
=> $\left(\frac{l}{2}\right)^2+b^2=2^2$
We know that the area of the rectangle is l*b, which can be obtained by considering 2 times the geometric mean of $\left(\frac{l}{2}\right)^2$ and $b^2$.
Therefore, for the maximum area, the equality condition of AM-GM inequality should be satisfied
=> $\left(\frac{l}{2}\right)^2=b^2$ => l = 2b.
=> l/b = 2/1.