The figure represents five concentric quarter-circles. The length of

Mind you, if you do have the options (as you will in GMAT), just plug in a value for x and solve. Working with numbers is much easier.
Radius of smallest circle is 1 and of largest is 5.

Required Area =\(\frac{1}{4}(\pi*25 - \pi*16 + \pi*9 - \pi*4 +\pi)\)
\(= 15\pi/4\)

In the options, just plug x = 5 and get your answer.
In the answer that Bunuel got above, if we plug x = 5, we get \(\frac{\pi(5^2 - 4*5 + 10)}{4} = 15\pi/4\)

The radii of the 5 quarter-circles are: x (the largest red), x-1 (the largest white), x-2 (next red), x-3 (next white), and x-4 (the inner red), so 5 radii for 5 quarter-circles.

We want the sum of 3 red (black) regions (so without two white regions in the middle of them).

The area of the largest red region is 1/4 of the are of the largest circle minus 1/4 of the area of the circle with radius of x-1 (the largest white circle) --> \(\frac{\pi{x^2}}{4}-\frac{\pi{(x-1)^2}}{4}=\frac{\pi}{4}(x^2-(x-1)^2)=\frac{\pi}{4}(2x-1)\);

The area of the next red region is 1/4 of the are of the circle with radius of x-2 (the next red circle) minus 1/4 of the area of the circle with radius of x-3 (the next white circle) --> \(\frac{\pi{(x-2)^2}}{4}-\frac{\pi{(x-3)^2}}{4}=\frac{\pi}{4}(2x-5)\);

Finally, the are of the red region in the center with the radius of (x-4) is \(\frac{\pi{(x-4)^2}}{4}=\frac{\pi}{4}(x^2-8x+16)\);

Sum: \(\frac{\pi}{4}(2x-1+2x-5+x^2-8x+16)=\frac{\pi}{4}(x^2-4x+10)\).
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The radius of the largest circle is x, so the radius of next is x-1 and so on so the area is
pi[(x^2-{x-1}^2)+({x-2}^2-{x-3}^2)+(x-4)^2]
on solving we get pi(x^2-4x+10)
since this is a quarter of a circle so the area is
1/4*pi(x^2-4x+10)

Using x=5, Area of shaded region

= \(\frac{1}{4} [π+(9π-4π)+(25π-16π)]\)
= \(\frac{15}{4}π\)

Hi Bunuel,

Would be great if you can add answer choices and OA to this question. You may add choices of your wish for the 4 incorrect choices in case the official ones are not available.

Warm Regards,
Pritishd

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Added the answer choices.
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I don't think you need to solve the whole thing.
\(\frac{𝜋}{4} [(x-4)^2 + (x-2)^2 - (x-3)^2 + (x)^2 - (x-1)^2]\)
From this the coefficient of \((x)^2\) would be \(1\). This means that we cannot take anything common from inside the bracket. So, \(\frac{𝜋}{4}\) will stay as it is and the correct option must contain \(\frac{𝜋}{4}\). So, C should be the answer.
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Hi bunuel please advise will it be safe to say that As we are asked combined area of shaded regions and it is given that region are quarter circle, we can assume correct option is third option c as it is only option which has π\4 for quarter

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Not necessarily. What if the expression following \(\frac{\pi}{4}\) was 4*something, 2*something, 1/2*something or 1/4*something?
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