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Frame Area

by dlencz » Mon Apr 23, 2012 11:04 am
The figure shows (not shown here) the top side of a circular medallion made of a circular piece of colored glass surrounded by a metal frame, represented by the shaded region. If the radius of the medallion is r centimeters and the width of the metal frame is s centimeters, then in terms of s and r, what is the area of the metal frame, in square centimeters?

(A) л(r - s)^2
(B) л(r^2 - s^2)
(C) 2л(r - s)
(D) лr(2r - s)
(E) лs(2r - s)

OA is E
Last edited by dlencz on Mon Apr 23, 2012 11:10 am, edited 1 time in total.
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by dlencz » Mon Apr 23, 2012 11:10 am
I plugged in numbers and subtracted the area of the medallion from the area of the medallion and frame, which should just leave the value of the frame.

R = 4
S = 1

Area of both: 5^2л = 25л
Area of medallion: 4^2л = 16л

25л - 16л = 9л

A is the equation that gives me 9л when I plug in the numbers so I can't see why it's wrong. Can someone please explain?
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by neelgandham » Mon Apr 23, 2012 12:26 pm
Radius of the medallion = r cm
Width of the frame = s cm
Radius of the colored glass = r-s cm (Look at the diagram in the attachment)
Area of medallion = pi*r*r square cms
Area of the colored glass = pi*(r-s)*(r-s)square cms

Shaded area = Area of the frame = Area of the medallion - Area of the colored glass
= pi*r*r - pi*(r-s)*(r-s)
= pi*(r*r - (r*r -r*s - r*s + s*s))
= pi*(r*r - r*r + r*s + r*s - s*s)
= pi*(2*r*s-s*s)
= Pi * s *(2*r-s)
Option E
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by GMATGuruNY » Mon Apr 23, 2012 5:41 pm
dlencz wrote:I plugged in numbers and subtracted the area of the medallion from the area of the medallion and frame, which should just leave the value of the frame.

R = 4
S = 1

Area of both: 5^2л = 25л
Area of medallion: 4^2л = 16л

25л - 16л = 9л

A is the equation that gives me 9л when I plug in the numbers so I can't see why it's wrong. Can someone please explain?
This question is easy to misread.

r = the radius of the ENTIRE MEDALLION (the circular glass center AND the frame).
If r=4 and s=1, then the radius of the circular glass center = 4-1 = 3.
Thus:
The area of the frame = the area of the entire medallion - the area of the circular center = 16л - 9л = 7л.

Now we plug r=4 and s=1 into the answers to see which yields an area of 7л.

Only answer choice E works:
лs(2r - s) = л(1)(2*4 - 1) = 7л.
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