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Wire cut into two pieces

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Wire cut into two pieces

by szDave » Thu Jan 24, 2013 6:19 am
Hello,

had trouble with this. Can you give me an explanation for why E is the right answer?

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 - 2(PI)r]^2
e) (PI)r^2 + [1O - (1/2)(PI)r]^2
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by Brent@GMATPrepNow » Thu Jan 24, 2013 6:37 am
szDave wrote: 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 - 2(PI)r]^2
e) (PI)r^2 + [1O - (1/2)(PI)r]^2
Here's one approach:

Since r is the radius of the circle, the area of the circle will be (pi)r^2.

If r is the radius of the circle, the length of wire used for this circle will equal its circumference which is 2(pi)r

So, the length of wire to be used for the square must equal 40 - 2(pi)r

In other words, the perimeter of the square will be 40 - 2(pi)r

Since squares have 4 equal sides, the length of each side of the square will be [40 - 2(pi)r]/4, which simplifies to be 10 - (pi)r/2

If each side of the square has length 10 - (pi)r/2, the area of the square will be [10 - (pi)r/2]^2

So, the total area will equal (pi)r^2 + [10 - (pi)r/2]^2, which is the same as E

Cheers,
Brent
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by GMATGuruNY » Thu Jan 24, 2013 7:44 am
szDave wrote:Hello,

had trouble with this. Can you give me an explanation for why E is the right answer?

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. πr²
B. πr² +10
C. πr² + 1/4 π²r²
D. πr² + (40-2πr)²
E. πr² + (10 - (1/2)πr)²
Let the ENTIRE WIRE be used to form the square.
Then:
Perimeter of the square = 40.
Side = 10.
Area = 100.

Circle area + square area = 0 + 100 = 100. This is our target.
Since the circle has no area, r=0.
Now we plug r=0 into the answers to see which yields our target of 100.
Only E works:
πr² + (10 - (1/2)πr)² = π0² + (10 - (1/2)π0)² = 0 + 10² = 100.

The correct answer is E.
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