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

Piece one = x ==> Circle

Piece two = 40 - x ==> Square

x = 2*pi*r (Since, piece one is circumference of circle)

40-x/4 => 10 - x/4 ==> 10 - 2*pi*r/4 (Since, piece two is perimeter of Square)

Total Area = Area of Circle + Area of square

=> pi * r^2 + (10 - pi*r/2)^2

[spoiler]{E}[/spoiler]

Zach.J.Dragone wrote: For starters, I had originally said "S" was the perimeter of the square and "40-S" was the perimeter of the circle which is technically correct, however, in the answer solution the perimeter of the square was 40-pi*d. I understand why that is, but I don't understand why we use that as opposed to something like "40-s" I need help figuring out the "why" for this question and others - the reasoning behind it.
Thanks!

"S" is a variable which is defined by you and is not present in the question.. So, you need to eliminate "S" from the answer.

Since, S is the circumference so.. S = 2*pi*r ..

Zach.J.Dragone 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 + (10 - 1/2*pi*r)^2

E

I have two problems - I understand the solution to this problem, but I am not sure I understand why.

For starters, I had originally said "S" was the perimeter of the square and "40-S" was the perimeter of the circle which is technically correct, however, in the answer solution the perimeter of the square was 40-pi*d. I understand why that is, but I don't understand why we use that as opposed to something like "40-s" I need help figuring out the "why" for this question and others - the reasoning behind it.

Thanks!

Let the wire be cut into 2 pieces of length x and y.
x + y = 40
The piece of length x is used to make a circle of radius r.
So the circumference of the circle = length x
2*pi*r = x

Now y = 40 - x
y = 40 - 2(pi)(r)
This is used to make a square.
The length of each side a = y/4 = (40 - 2(pi)(r))/4
a = 10 - pi*r/2

Area of the circle = pi * r^2
Area of the square = (10 - pi*r/2)^2

So, the total area in terms of r = pi*r^2 + (10 - 1/2*pi*r)^2

Choose E

Cheers

GMAT is trying to trap you wen you write a equation for this question. easier way out is you do not write equation for solving this problem. This is a flexible problem you need to realise first. any value of r will satisfy the final equation that we will come up with . so how about assuming r to be zero !!! this means i make only a square and no circle at all. all four side of a square is equal so this square has a side of 40/4 = 10 . now area of the square = 100.

now the answers are given in terms of variable r . when we started solving this question we assumed r = 0 . so put r = 0 in your answer choice and if you dont get 100 then rule that answer option out.

(a) pi r^2 = pi . 0 = 0 .... (not equal to 100 so rule out this option)
(b)pi*r^2 + 10 = 10 (rule this out as not equal to 100)
(c) pi*r^2 + 1/4*pi^2*r^2= 0 rule out
(d) pi*r^2 + (40 - 2\pi*r)^2 = 1600 ... rule out
(e) pi*r^2 + (10 - 1/2*pi*r)^2 = 100 ... matches 100 .. so the answer ..



One more similar question from G prep itself - just for practice - (felxible ones )


Before being simplified, the instructions for computing income tax in country R were to add 2 percent of one's annual income to the average (arithmetic mean) of 100 units of country R's currency and 1 percent of one's annual income. Which of the following represents the simplified formula for computing the income tax in country R's currency, for a person in that country whose annual income is I?
a) 50 + (I/200)
b) 50 + (3I/100)
c) 50 + (I/40)
d) 100 + (I/50)
e) 100 + (3I/100)

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

Since the answer choices are in terms of a variable -- the value of r -- we can PLUG IN.

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.

GMATGuruNY 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. πr²
B. πr² +10
C. πr² + 1/4 π²r²
D. πr² + (40-2πr)²
E. πr² + (10 - (1/2)πr)²

Since the answer choices are in terms of a variable -- the value of r -- we can PLUG IN.

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.

Hi Guru,

Can you please tell me why did you take r=0, but not any value? Is it because of calculation?

Thanks
Nandish

Zach.J.Dragone 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 + (10 - 1/2*pi*r)^2

We are given that 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.

Since the circumference of a circle with radius r is 2πr, the amount of wire used to form the circle is 2πr. Thus, we have 40 – 2πr left over to form the square. In other words, the perimeter of the square is 40 – 2πr. However, since we need to calculate the total area of the circular and the square regions, we need to determine the side of the square in terms of r. Since the perimeter of the square is 40 – 2πr, the side of the square is:

side = (40 – 2πr)/4

side = 10 – (1/2)πr

Now we can determine the areas of the circle and the square.

Area of circle = πr2

Area of square = side^2 = (10 – (1/2)πr)^2

Thus, the combined area of the circle and square is πr2 + (10 – (1/2)πr)^2.

Answer: E