In the given figure, ABCD is a rectangle (AB BC) and BC and AD are

Question

https://gmatclub.com/forum/download/file.php?id=71428 In the given figure, ABCD is a rectangle (AB > BC) and BC and AD are the diameters of the semi-circle. If the perimeter of the figure, excluding the length BC and AD is 86 inches and the length, breadth, and radius are integers, what is the ratio of the area of the ABCD and the area of the two semi-circles? (π = \frac{22}{7} ) A. 10:11 B. 11:10 C. 20:11 D. 30:21 E. None of the above Image - 3.png

gmatophobia · Answer

Perimeter = AB + Arc AD + DC + Arc CB

Assume that the radius of the semi-circle = x

AD = BC = 2x
AB = DC = y

2y + 2*\pi * x = 86

2y + 2* \frac{22}{7}* x = 86

Dividing both sides by 2

y + \frac{22}{7}* x = 43

7y + 22x = 43*7

7y = 43*7 - 21x - x

y = 43 - 3x - \frac{x}{7}

x = 0; y = 43

Other values are (x,y) = (7,21)

Ratio of the area of the ABCD and the area of the two semi-circles =  \frac{14*21}{\pi * 7* 7}

=  \frac{14*21}{\pi * 7 * 7 }

=  \frac{14*21*7}{22 * 7 * 7 }

=  \frac{21}{11 }

Option E

In the given figure, ABCD is a rectangle (AB > BC) and BC and AD are

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