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CAT 2018 Slot 2 — QA Question 19

Basics of Mensuration/PrismEasy

From a rectangle ABCD of area 768 sqcm, a semicircular part with diameter AB and area 72π sqcm is removed. The perimeter of the leftover portion, in cm, is:

Answer & solution

Correct answer: 88 + 12π

  • 88 + 12π

  • B

    82 + 24π

  • C

    80 + 16π

  • D

    86 + 8π

Solution

Easy

From the semicircle's area get its radius and hence ABAB. The rectangle's area then gives the other side. The leftover boundary is three full sides of the rectangle plus the curved arc (the side ABAB is replaced by the semicircular arc).

1

Radius and diameter from the semicircle. A semicircle of radius rr has area 12πr2\tfrac12\pi r^2.

πr22=72π⇒ r2=144⇒r=12⇒ AB=2r=24\begin{aligned} &\frac{\pi r^2}{2}=72\pi\\ &\Rightarrow\ r^2=144\Rightarrow r=12\\ &\Rightarrow\ AB=2r=24 \end{aligned}
2

Other side of the rectangle. Area =768=768 with one side AB=24AB=24.

other side=76824=32\begin{aligned} &\text{other side}=\frac{768}{24}=32 \end{aligned}
3

Add up the leftover boundary. Side ABAB is removed and replaced by the semicircular arc of length πr=12π\pi r=12\pi. The other three sides (3232, 2424, 3232) remain.

perimeter=πr+32+24+32=12π+88⇒ 88+12π\begin{aligned} &\text{perimeter}=\pi r+32+24+32\\ &=12\pi+88\\ &\Rightarrow\ 88+12\pi \end{aligned}
A B C D 32 32 24 arc = 12π
perimeter=88+12π cm\text{perimeter}=88+12\pi\text{ cm}

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CAT 2018 Slot 2 QA Q19: From a rectangle ABCD of area 768 sqcm, a semicircular part with diameter AB and area 72π sqcm is removed. — Solution | TheCATExam