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CAT 2020 Slot 1 — QA Question 14

Basics of Mensuration/PrismEasy

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

Answer & solution

  • A

    3π(52+6π)

  • B

    4π(3+9π)

  • 3π(5+12π)

  • D

    5π(3+9π)

Solution

Easy

Unpainted area is 23\tfrac23 of the painted (circle) area, so the circle is 35\tfrac35 of the whole sheet. Since the circle touches the two opposite sides, the rectangle's width equals the diameter 2r2r. Find rr from the circle's area, then the length from the total area, and assemble the perimeter as 2(length+width)2(\text{length}+\text{width}) with width =2r=2r.

r width = 2r (circle touches top & bottom) length = l
1

Fraction of the sheet that is painted. Let painted area be PP; unpainted =23P=\tfrac23 P, and these sum to the whole sheet.

P+23P=135⇒ 53P=135⇒ P=81(area of the circle)\begin{aligned} &P+\tfrac23 P = 135\\ &\Rightarrow\ \tfrac53 P = 135\\ &\Rightarrow\ P = 81 \quad\text{(area of the circle)} \end{aligned}
2

Radius of the circle.

πr2=81⇒ r=9π\begin{aligned} &\pi r^2 = 81\\ &\Rightarrow\ r = \frac{9}{\sqrt{\pi}} \end{aligned}
3

Length of the rectangle. The circle touches the two opposite sides, so the width =2r=2r, and the total area gives the length.

135=(width)×l=2r⋅l⇒ l=1352r=1352⋅9π=15π2\begin{aligned} &135 = (\text{width})\times l = 2r\cdot l\\ &\Rightarrow\ l = \frac{135}{2r} = \frac{135}{2\cdot \tfrac{9}{\sqrt{\pi}}} = \frac{15\sqrt{\pi}}{2} \end{aligned}
4

Perimeter. Length is ll and width is 2r2r, so the perimeter is 2(length+width)=2l+2(2r)2(\text{length}+\text{width}) = 2l + 2(2r).

Perimeter=2l+4r=2⋅15π2+4⋅9π=15π+36π=3π(5+12π)(factor out 3π)\begin{aligned} &\text{Perimeter} = 2l + 4r\\ &= 2\cdot\frac{15\sqrt{\pi}}{2} + 4\cdot\frac{9}{\sqrt{\pi}}\\ &= 15\sqrt{\pi} + \frac{36}{\sqrt{\pi}}\\ &= 3\sqrt{\pi}\left(5 + \frac{12}{\pi}\right) \quad\text{(factor out }3\sqrt{\pi}\text{)} \end{aligned}
3π(5+12π)— option (c)3\sqrt{\pi}\left(5 + \frac{12}{\pi}\right)\quad\text{— option (c)}
CAT 2020 Slot 1 QA Q14: On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite — Solution | TheCATExam