GMATPrep 1 Geometry Question

[Mjkourtis]

A thin wire of 40 meters 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 represents the total area?
a. πr^2
b. πr^2 +10
c. πr^2 + (1/4)*π^2*r^2
d. πr^2 + (40- 2πr)^2
e. πr^2 + (10-(1/2)*πr)^2

[GMATGuruNY]

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 r=0, so that the ENTIRE wire is used to form the square.
Perimeter of the square = 40.
Each side of the square = 40/4 = 10.
Area of the square = 10² = 100.
Area of the circle + area of the square = 0 + 100 = 100. This is our target.

Now we plug r=0 into the answers to see which yields our target of 100.

A quick scan reveals that only E works:
πr² + (10 - 1/2πr)² = 0 + 10² = 100.

The correct answer is E.

[anuprajan5]

We can find the answer as such.

There are 2 parts of the wire.

x and (40-x)

Assume x as the circumference of the circle with radius r.

x = 2pi*r



Since 40-x forms the perimeter of the square, side a = (40-x)/4

a = 10-x/4

Area = a^2 = (10-(x/4))^2

Substituting for x = (10-(2pi*r)/4)^2 = (10-pi*r/2)^2


Total area = Area of circle + Are of square

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

Answer E