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by gvosough » Mon Feb 14, 2011 5:02 pm
A thin piece of wire 40m long is cut into two pieces. one piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. What's the total area in sq meters of the circlar and the square regions in terms of r?

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by GMATGuruNY » Mon Feb 14, 2011 5:24 pm
A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and the square regions in terms of r?

A. pi r^2
B. pi r^2 +10
C. pi r^2 + 1/4 pi^2 r^2
D. pi r^2 + (40-2pi r)^2
E. pi r^2 + (10 - 1/2 pi r)^2
Plug in r = 4.
Area of circle = �r² = �4² = 16� ≈ 48
Circumference of circle = 2�r = 8� ≈ 24.
Perimeter of square = remaining wire = 40-24 = 16.
Side of square = 16/4 = 4.
Area of square = 4^2 = 16.
Area of circle + area of square ≈ 48+16 ≈ 64. This is our target.

Now we plug r=4 into all the answers to see which yields our target of 64.

Only answer choice E works:
pi r^2 + (10 - 1/2 pi r)^2 ≈ 3*(4²) + (10 - 1/2*3*4)² = 48+16 = 64.

The correct answer is E.
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by Anurag@Gurome » Mon Feb 14, 2011 7:19 pm
gvosough wrote:A thin piece of wire 40m long is cut into two pieces. one piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. What's the total area in sq meters of the circlar and the square regions in terms of r?

Thanks.
Another approach:
The length of wire is 40m.
Let the length of pieces into which it is cut be x and 40-x.
Let the piece with length x be used to form a circle with radius r.
Or 2*pi*r = x.
Also, area of circle is pi * r^2.
The piece with length 40-x is used to make a square.
Let the side of square be s.
Or s = (40-x)/4.
Or area of square is s^2 = {(40-x)/4}^2 = (10 – 2*pi*r/4)^2=(10 – pi*r/2)^2.
Or area of circle and the square is pi*r^2 + (10 – pi*r/2)^2.
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