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Circular Ring With Thickness

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by EOV » Mon Dec 03, 2012 7:54 am
Hi everybody,

Image

The figure here is a ring with a thickness shown in black. If the radius of the ring is r and the thickness of the ring is t, then which of the following expressions best describes the area of the ring?

(A) π(r−t)2
(B) π(r2−t2)
(C) 2π(r−2t)
(D) πr(2r−t)
(E) πt(2r−t)

Could you please help me to undestand correct answer?
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by eaakbari » Mon Dec 03, 2012 12:19 pm
IMO E.

The question is vague as it does not explicitly mention whether r is the internal radius or the external.

Assuming that r is external radius.
Radius of empty hole space = r-t

Area of Ring will be

Area of full circle - Area of empty hole

π r^2 - π (r-t)^2

which simplifies to

Ï€ t(2r-t)

Hence E
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by nisagl750 » Thu Dec 06, 2012 4:45 am
EOV wrote:Hi everybody,

Image

The figure here is a ring with a thickness shown in black. If the radius of the ring is r and the thickness of the ring is t, then which of the following expressions best describes the area of the ring?

(A) π(r−t)2
(B) π(r2−t2)
(C) 2π(r−2t)
(D) πr(2r−t)
(E) πt(2r−t)

Could you please help me to undestand correct answer?
Consider the ring as two concentric circles.

Image

Since it is not given in question whether r is the radius of the outer circle or inner circle, Let's try both the cases...

Case:1
The radius of inner circle is r
The radius of outer circle is (r+t)
Area of the Ring = Area of outer circle - Area of inner circle
Ï€(r+t)^2-Ï€r^2
= πr^2 + πt^2 + 2πrt - πr^2
= πt^2 + 2πrt
= πt(t+2r)
Which is not in any of the options.....
So r is the radius of Outer circle

Case:2
The radius of outer circle is r
The radius of inner circle is (r-t)
Area of the Ring = Area of outer circle - Area of inner circle
Ï€r^2-Ï€(r-t)^2
= πr^2 - πr^2 -πt^2 + 2πrt
= 2Ï€rt-Ï€t^2
= πt(2r-t)
Option E
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