Question

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

Answer 1

Answer: choice E

total area = pi*r^2 + (10 - (1/2)pi*r)^2

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How I got that answer:

Draw out a line segment and label it to be 40 meters long. Somewhere in the middle (not the exact midpoint necessarily), draw a cutting line to divide the segment into two (not necessarily equal) parts. The first part is x units long. The remaining bit is 40-x units long. See the attached image below. The drawing is optional, but it might help to visually see how the two parts interact.

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The wire that is x units long is used to form a circle of radius r. The question is: what is x in terms of r?

The circumference of the circle of radius r is

C = 2*pi*r

If we cut the circle and unroll it out into a straight line, then its perimeter or circumference will stretch into a line of length 2*pi*r. So this must be the length of x.

x = 2*pi*r

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The value of 40-x is then

40-x = 40-2pi*r

which will be used to form the square.

The total perimeter of the square is P = 4*s, where s is the side length. Solving for s gets to

s = P/4

Plug in P = (40-2pi*r) and simplify

s = P/4

s = (40-2pi*r)/4

s = 40/4-(2pi*r)/4

s = 40/4-(2/4)pi*r

s = 10 - (1/2)pi*r

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We know the side length of the square, which can be used to find the area of the square

A = area of square

A = s^2

A = (10-(1/2)*pi*r)^2

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M = area of circle = pi*r^2

N = area of square = (10-(1/2)*pi*r)^2

total area = M+N

total area = pi*r^2 + (10-(1/2)*pi*r)^2