CAT 2020 Slot 1 QA Q22 Mensuration

Question 22 of the Quantitative Ability (QA) section, CAT 2020 Slot 1, with CAT's official answer and a worked solution.

On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches opposite two sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

A
3√π(5 + \(\frac{12}{π}\))
B
4√π(3 + \(\frac{12}{π}\))
C
5√π(3 + \(\frac{12}{π}\))
D
3√π(\(\frac{5}{2}\) + \(\frac{6}{π}\))

Answer: (A) 3√π(5 + 12/(π) ) Mensuration MCQ

Solution

The circle touches two opposite sides, so the width is 2r. Unpainted \(= \frac23\) painted, so the sheet \(= \frac53\pi r^2 = 135\), giving \(\pi r^2 = 81\) and \(r = \frac{9}{\sqrt\pi}\).
Width \(= \frac{18}{\sqrt\pi}\), length \(= \frac{135\sqrt\pi}{18} = 7.5\sqrt\pi\).
Perimeter \(= 15\sqrt\pi + \frac{36}{\sqrt\pi} = 3\sqrt\pi\left(5+\frac{12}{\pi}\right)\).