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# What is the area of the shaded region, if the circumference of the lar

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Math Expert
Joined: 02 Sep 2009
Posts: 53060
What is the area of the shaded region, if the circumference of the lar  [#permalink]

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02 Sep 2018, 22:31
00:00

Difficulty:

15% (low)

Question Stats:

91% (01:35) correct 9% (01:29) wrong based on 51 sessions

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What is the area of the shaded region, if the circumference of the larger circle is 10π and the center of the larger circle lies on the small circle, as shown?

A. 25π
B. 22.75π
C. 18.75π
D. 6.25π
E. 2.25π

Attachment:

image034.jpg [ 3.85 KiB | Viewed 759 times ]

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Re: What is the area of the shaded region, if the circumference of the lar  [#permalink]

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02 Sep 2018, 23:57
Bunuel wrote:

What is the area of the shaded region, if the circumference of the larger circle is 10π and the center of the larger circle lies on the small circle, as shown?

A. 25π
B. 22.75π
C. 18.75π
D. 6.25π
E. 2.25π

Attachment:
image034.jpg

Circumference of larger circle = 2πR = 10π

i.e. Radius of smaller circle = 5/2

Area of shaded region = Bigger circle - smaller circle = π(5^2 - 2.5^2) = 18.75π

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Re: What is the area of the shaded region, if the circumference of the lar  [#permalink]

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03 Sep 2018, 06:56
Bunuel wrote:

What is the area of the shaded region, if the circumference of the larger circle is 10π and the center of the larger circle lies on the small circle, as shown?

A. 25π
B. 22.75π
C. 18.75π
D. 6.25π
E. 2.25π

Attachment:
image034.jpg

$$2πR = 10π$$ {R = RAdius of the Bigger Circle }

So, $$R = 5$$

Now, $$r = \frac{5}{2} = 2.5$$ { r = Radius of the smaller circle }

So, The Difference of the Area is $$π ( R^2 - r^2) = π ( R + r ) ( R - r )$$

Or, The required area is $$π * 7.5 * 2.5 = 18.75π$$ , Answer must be (C)
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Re: What is the area of the shaded region, if the circumference of the lar  [#permalink]

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03 Sep 2018, 09:12
Circumference of the larger circle = 2πr

2πr = 10
πr = 10/2
r = 5

Area of the larger circle = πr^2
= 5^2 x π
= 25π

Area of the smaller circle = πr^2
= 2.5^2 x π
= 6.25π

Area of the shaded region = 25π - 6.25π
= 18.75π

Hence in my opinion C
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Re: What is the area of the shaded region, if the circumference of the lar  [#permalink]

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07 Sep 2018, 16:08
Bunuel wrote:

What is the area of the shaded region, if the circumference of the larger circle is 10π and the center of the larger circle lies on the small circle, as shown?

A. 25π
B. 22.75π
C. 18.75π
D. 6.25π
E. 2.25π

Attachment:
image034.jpg

Since the circumference of the larger circle is 10π, the diameter is 10, and the radius is 5.

Thus, the diameter of the smaller circle is 5, and its radius is 2.5.

So, the area of the larger circle is 5^2 x π = 25π and the area of the smaller circle is 2.5^2 x π = 6.25π.

Therefore, the area of the shaded region is 25π - 6.25π = 18.75π.

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Re: What is the area of the shaded region, if the circumference of the lar   [#permalink] 07 Sep 2018, 16:08
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