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# In the figure above, an unshaded circle with diameter x is s

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Founder
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In the figure above, an unshaded circle with diameter x is s [#permalink]  25 May 2020, 10:36
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Question Stats:

44% (01:51) correct 55% (03:16) wrong based on 18 sessions

In the figure above, an unshaded circle with diameter x is surrounded by a shaded region with uniform width of 2. For what values of x is the area of the unshaded circle greater than the area of the shaded region?

Indicate all such values.

❑ 6

❑ 7

❑ 8

❑ 9

❑ 10

❑ 11

[Reveal] Spoiler:
Attachment:

#greprepclub In the figure above, an unshaded circle with diameter x.jpg [ 14.39 KiB | Viewed 610 times ]
[Reveal] Spoiler: OA

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Founder
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Re: In the figure above, an unshaded circle with diameter x is s [#permalink]  25 May 2020, 10:37
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Post A Detailed Correct Solution For The Above Questions And Get A Kudos.
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Re: In the figure above, an unshaded circle with diameter x is s [#permalink]  28 May 2020, 05:11
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Radius of unshaded circle = $$\frac{x}{2}$$

Area of unshaded circle = $$\pi$$ * $$(\frac{x}{2})^2$$

Area of shaded + unshaded circle = $$\pi$$ * $$(\frac{x}{2 }+2 )^2$$

Area of shaded circle = $$\pi$$ * $$(\frac{x}{2 }+2 )^2$$ - $$\pi$$ * $$(\frac{x}{2})^2$$

if area of unshaded circle is greater than area of shaded circle then

$$\pi$$ * $$(\frac{x}{2})^2$$ > $$\pi$$ * $$(\frac{x}{2 }+2 )^2$$ - $$\pi$$ * $$(\frac{x}{2})^2$$

Equation simplifies to $$(\frac{x}{2})^2$$ > $$(\frac{x}{2 }+2 )^2$$ - $$(\frac{x}{2})^2$$

$$\frac{x^2 }{4}$$ > $$\frac{x^2 }{4}$$ + 4 + 2x - $$\frac{x^2 }{4}$$

$$x^2$$ > 16 + 8x

Now substituting all values of x in answer choices we see that only x = 10 and x=11 satisfies above equation for x

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Re: In the figure above, an unshaded circle with diameter x is s [#permalink]  13 Jun 2020, 12:55
Any other explanation by taking diameter instead of radius? or any simple explanation plz?
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Re: In the figure above, an unshaded circle with diameter x is s [#permalink]  14 Jun 2020, 07:25
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Expert's post
Farina wrote:
Any other explanation by taking diameter instead of radius? or any simple explanation plz?

Since the area formula (area $$= \pi r^2$$) it would only complicate matters to try to use the diameter.
It makes much more sense to answer the question in terms of radius and then convert those results to diameters.
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Re: In the figure above, an unshaded circle with diameter x is s   [#permalink] 14 Jun 2020, 07:25
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