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40 meter wire

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40 meter wire

by msbelasco » Mon Aug 06, 2012 7:43 pm
A thin piece of wire 40 meters long is cut into 2 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 circularand the square regions in terms of r?

Ï€r^2

Ï€r^2 + 10

πr^2 + 1/4 π^2 r^2

Ï€r^2 + (40 - 2Ï€r)^2

Ï€r^2 + (10 -1/2Ï€r)^2

Thanks for any help.
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by Anurag@Gurome » Mon Aug 06, 2012 7:45 pm
msbelasco wrote:A thin piece of wire 40 meters long is cut into 2 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 circularand the square regions in terms of r?

Ï€r^2

Ï€r^2 + 10

πr^2 + 1/4 π^2 r^2

Ï€r^2 + (40 - 2Ï€r)^2

Ï€r^2 + (10 -1/2Ï€r)^2

Thanks for any help.
The length of one piece = 2(pi)r and the other is a square. Let us assume that each side of the square is "a".
Perimeter of square = 4a
Now, 2(pi)r + 4a = 40
4a = 40 - 2(pi)r
a = [40 - 2(pi)r]/4 = 10 - (pi)r/2

Total area = Area of Circle + Area of Square
= (pi)r² + a²
= [spoiler](pi)r² + (10 - (pi)r/2)² [/spoiler]

The correct answer is E.
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by armand_h » Tue Aug 07, 2012 2:21 am
you can do some checks before you start solving equations:
for example a very quick check is to set r to zero and check the results for each suggested solution. We know that with r=0 the to total area will be the are of the square, which in this case is(40/4)^2=100

You will notice very quickly that the solution is [spoiler]Ï€r^2 + (10 -1/2Ï€r)^2 [/spoiler]
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