A thin piece of wire 40 metres long is cut into two pieces.

[GMAT 5/18]

A thin piece of wire 40 metres long is cut into two pieces. One piece is used to form a circle with radius r , and the other is used to form square. No wire if 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 (Formula for an area of a circle)
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 + (10 - 0.5.Pi.r)^2

The way I did this question was to denote each piece 20m long. This made a square with sides of 5m and a circle with a radius of 20m. So, the total area I got was Pi.400 + 25 (metres squared). The correct answer was e., and I guessed e, but that answer does not equal my total area.

Thanks!
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[Guest]

The answer shld be E

2.pi.r+4s(side of the sqare) = 40
s=40-2.pi.r)/4==> (10-.5pi.r)^2

Therefore
Total area = Area of the circle (pi.r^2)+(10-.5pi.)^2 ==> E

[GMAT 5/18]

Ah, I see now where I made my mistake. The 20 I used is not the radius, but rather the circumference of the circle. So r = 10/Pi.

Now, my total area = Pi.r^2 + 25, and answer e. works!

Thanks a lot!

[Guest]

How would you complete this problem without picking numbers? Or, is picking numbers the way to go for this problem? If so, what flags should I have seen to alert me that this is a picking number type problem.

[Kunal]

This is how to do without the using actual numbers.

Area of circle = PI.r^2
side of the square = 40-2.PI.r/4 = 10 - 0.5.PI.r
Area of square = (10 - 0.5.PI.r)^2 =

Add both the areas above. The answer works out to E.

[RonPurewal]

so you have a total length of 40.

you use part of that length to make a circle with radius r. this circle will have circumference 2πr, so the leftover amount with which to make the square is 40 - 2πr.

therefore, each side of the square is one-fourth of (40 - 2πr), or 10 - 0.5πr.
the area of the square is thus (10 - 0.5πr)^2, so the answer (e) follows.

--

by the way, i would not recommend number picking on this problem, as the formulas are terribly awkward - not to mention the fact that you actually don't have to do that much work to derive the formula. the hard part is getting past the conceptual barrier of realizing that you have to subtract the circle's circumference from 40 in order to find out the amount of wire left to make the square.

[benkriger]

I chose to cut the rope into two parts. I chose 8 and 32, keeping in mine one number had to be divisible by 4 (for the square).

If I used the 32 inch piece for the square, that means it has sides of 8. And an area of 64.

Looking at the answer choices, I realize I only need to calculate the area of the square, because the area of the circle is already there. (If we know that for the circle C=8, then we know we can find R, and then we know we can find Area).

Starting with choice E, I see that the math for the second part comes out to 64.

90% certain, I selected E and moved on, as I pretty much reach the point of going over the time limit.

Hope that helps a little....

[tim]

interesting..

[TokuS705]

Hi guys, so I totally understand how to get to the answer E but when I originally did this question I got to answer C. Here is the calculation that I went through to get to C and I don't understand what's wrong with it.

First, I divided the 40 meters into 20 meters each.
Next, I said 2*Pi*r = 20 for the circle and 4s = 20 for the square.
Next, I just kept the r and said Pi*r^2 is the area of the circle.
Now to figure out the area of the square, I figured out that s = 5. Taking the previous equation for the circumference of the circle, I determined that Pi*r/2 = 5. This is also of course the length of the side of the square.
When you square this you get Pi^2*r^2/4. This should also equal the area of the square.
Add the two together, you get Pi*r^2 + Pi^2*r^2/4. This is C.

What am I doing wrong?

[RonPurewal]

First, I divided the 40 meters into 20 meters each

that ^^ is the problem.

the correct answer has to work for EVERY possible value, not just one. (also, if you're going to pick "smart numbers", you should definitely avoid values like this! the more symmetrical the values, the more likely you'll be to have a coincidence in the answer choices -- exactly what happened here)

[mondegreen]

You could assume that the value of r=0, i.e the entire 40 mts is used to form a square. Thus, the total perimter of the square = 40
or 4*(side of square) = 40
or side of square = 10.
Hence, the area of the square = 100 sq. mts
Just put r=0 in the options, only option E satisfies.

You could also make the perimeter of the square zero, but that means unnecessary pi.

[RonPurewal]

^^ that works too.