Problem Solving

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!

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