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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Karishma
Veritas Prep GMAT Instructor
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