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CAT 2003 Slot 1 — QA Question 32

2 CirclesEasy

Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the semi-circular field with centre at O. Horses are tied up at P, R and S such that PO and RO are the radii of semi-circles with centres at P and R respectively, and S is the centre of the circle touching the two semi-circles with diameters AO and OB. The horses tied at P and R can graze within the respective semi-circles and the horse tied at S can graze within the circle centred at S. The percentage of the area of the semi-circle with diameter AB that cannot be grazed by the horses is nearest to

Answer & solution

Correct answer: 28

  • A

    20

  • 28

  • C

    36

  • D

    40

Solution

Consider the diagram below.

Let r be the radius of the smaller semi-circles and s be the radius of the smaller circle.

OS = 2r − s

PS = r + s

PO = r

But, ∆PSO is a right-angled triangle.

∴ PS2 = PO2 + SO2

(r + s)2 = r2 + (2r − s)2

∴ r² + s² + 2rs = r² + 4r² + s² − 4rs

 ∴s=2r3

∴ Total area not grazed = 12 π(2r)2 - [2×12πr2+π(2r3]2)

= 2πr2 - πr2 - 49πr2

= 59πr2

∴ Required percentage = 59πr22πr2 × 100

= 59×2 × 100

≈ 28%

Hence, option (b).

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CAT 2003 Slot 1 QA Q32: Three horses are grazing within a semi-circular field. In the diagram given below, AB is the diameter of the s — Solution | TheCATExam