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The figure represents five concentric quarter-circles. The

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The figure represents five concentric quarter-circles. The  [#permalink]

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New post 08 Feb 2012, 17:13
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The figure represents five concentric quarter-circles. The length of the radius of the largest quarter-circle is x. The length of the radius of each successively smaller quarter-circle is one less than that of the next larger quarter-circle. What is the combined area of the shaded regions (black), in terms of x?
Attachment:
Circles+.PNG
Circles+.PNG [ 2.4 KiB | Viewed 9278 times ]



Guys - I neither have answer choices nor the OA. Can someone please help? This is how I am trying to solve it.
Area of the larger circle = pi \(x^2\)

Area of middle circle = pi (x-1)\(^2\)

Area of outer circle = pi (x-2)\(^2\)

Combined area should be the sum of the above 3. So when I do the maths I get combined area as

pi (3\(x^2\) - 4x +5). But I am stuck after this.
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Re: The figure represents five concentric quarter-circles. The  [#permalink]

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New post 09 Feb 2012, 02:27
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enigma123 wrote:
The figure represents five concentric quarter-circles. The length of the radius of the largest quarter-circle is x. The length of the radius of each successively smaller quarter-circle is one less than that of the next larger quarter-circle. What is the combined area of the shaded regions (black), in terms of x?
Attachment:
Circles+.PNG



Guys - I neither have answer choices nor the OA. Can someone please help? This is how I am trying to solve it.
Area of the larger circle = pi \(x^2\)

Area of middle circle = pi (x-1)\(^2\)

Area of outer circle = pi (x-2)\(^2\)

Combined area should be the sum of the above 3. So when I do the maths I get combined area as

pi (3\(x^2\) - 4x +5). But I am stuck after this.


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 \(\frac{\pi(5^2 - 4*5 + 10)}{4} = 15\pi/4\)
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Re: Combined areas of circles  [#permalink]

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New post 08 Feb 2012, 17:46
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enigma123 wrote:
The figure represents five concentric quarter-circles. The length of the radius of the largest quarter-circle is x. The length of the radius of each successively smaller quarter-circle is one less than that of the next larger quarter-circle. What is the combined area of the shaded regions (black), in terms of x?
Attachment:
circles.PNG
circles.PNG [ 25.38 KiB | Viewed 9204 times ]


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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Re: The figure represents five concentric quarter-circles. The  [#permalink]

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New post 09 Feb 2012, 07:19
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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)
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Re: The figure represents five concentric quarter-circles. The  [#permalink]

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New post 06 Jul 2017, 13:54
Using x=5, Area of shaded region

= \(\frac{1}{4} [π+(9π-4π)+(25π-16π)]\)
= \(\frac{15}{4}π\)
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Re: The figure represents five concentric quarter-circles. The  [#permalink]

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