CAT 2020 Slot 1 QA Question & Solution
GeometryMedium
Question
On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite 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
Options
$3\sqrt{\pi}(5+\frac{12}{\pi})$
$4\sqrt{\pi}(3+\frac{9}{\pi})$
$3\sqrt{\pi}(\frac{5}{2}+\frac{6}{\pi})$
$5\sqrt{\pi}(3+\frac{9}{\pi})$
Solution
Let ABCD be the rectangle with length 2l and breadth 2b respectively.
Area of the circle i.e. area of painted region = $\pi\ b^2$.
Given, 4lb-$\pi\ b^2$=(2/3)$\pi\ b^2$.
=> 4lb=(5/3)$\pi\ b^2$.
=>l=$\frac{5\pi}{12}b$.
Given, 4lb=135 => 4*$\frac{5\pi}{12}b^2$=135 => b= $\frac{9}{\sqrt{\ \pi\ }}$
=> l=$\frac{15}{4}\sqrt{\ \pi\ }$
Perimeter of rectangle =4(l+b)=4($\frac{15}{4}\sqrt{\ \pi\ }$+$\frac{9}{\sqrt{\ \pi\ }}$ )=$3\sqrt{\pi}(5+\frac{12}{\pi})$.
Hence option A is correct.