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03 Jul 2018, 09:25
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (03:37) correct
63%
(02:46)
wrong
based on 43
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ABCD is a square and AEC and AFC are one fourth of the circumference of the circle whose radius is equal to the length of the side of the square ABCD. Ratio of unshaded to shaded region? image is attached below
(A) 22 : 3
(B) 4 : 3
(C) 3 : 4
(D) 7 : 4
(E) cannot be determined
Attachment:
Image50.gif [ 2.19 KiB | Viewed 3663 times ]
03 Jul 2018, 12:44
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OA: C
Let side of Square = \(a\)
Area of Shaded portion = \(2*\frac{\pi*a^2}{4}-a^2\)\(=\) \(\frac{\pi*a^2}{2}-a^2 =\frac{\pi*a^2-2*a^2}{2}\)
Area of Unshaded portion = Area of Square - Area of Shaded portion
= \(a^2 - (2*\frac{\pi*a^2}{4}-a^2)\)\(=\) \(2*a^2 - \frac{\pi*a^2}{2}=\frac{4*a^2 - \pi*a^2}{2}\)
\(\frac{Area of Unshaded portion}{Area of Shaded portion}=\frac{4*a^2 - \pi*a^2}{\pi*a^2-2*a^2}=\frac{4 - \pi}{\pi-2}\)
Putting \(\pi = \frac{22}{7}\) in above ratio , we get \(\frac{Area of Unshaded portion}{Area of Shaded portion}= \frac{(28-22)}{(22-14)} =\frac{6}{8}=\frac{3}{4}\)
Let side of Square = \(a\)
Area of Shaded portion = \(2*\frac{\pi*a^2}{4}-a^2\)\(=\) \(\frac{\pi*a^2}{2}-a^2 =\frac{\pi*a^2-2*a^2}{2}\)
Area of Unshaded portion = Area of Square - Area of Shaded portion
= \(a^2 - (2*\frac{\pi*a^2}{4}-a^2)\)\(=\) \(2*a^2 - \frac{\pi*a^2}{2}=\frac{4*a^2 - \pi*a^2}{2}\)
\(\frac{Area of Unshaded portion}{Area of Shaded portion}=\frac{4*a^2 - \pi*a^2}{\pi*a^2-2*a^2}=\frac{4 - \pi}{\pi-2}\)
Putting \(\pi = \frac{22}{7}\) in above ratio , we get \(\frac{Area of Unshaded portion}{Area of Shaded portion}= \frac{(28-22)}{(22-14)} =\frac{6}{8}=\frac{3}{4}\)
03 Jul 2018, 13:03
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Let's assume the side of the square to be 7. The area of this square is 49.
Similarly, as the length of the sector is also 7, the area of one sector \(\frac{1}{4}*\frac{22}{7}*7^2 = \frac{77}{2}\)
There are 2 such sectors AEC and AFC each having the same area, making the total area \(77\)
We use the following diagram to find the area of the shaded region
Attachment:
Image50.gif
Area of AEC = 1 + 2 | Area of AFC = 2 + 3 | Area of Square = 1 + 2 + 3
The area of shaded region = Area of AEC + Area of AFC - Area of Square = 77 - 49 = 28
Area of unshaded region = Area of square - Area of shaded region = 49 - 28 = 21
Therefore, the ratio of the unshaded to the shaded region is 21:28 or 3:4(Option C)
