<h3>Answer: choice E</h3><h3>total area = pi*r^2 + (10 - (1/2)pi*r)^2</h3>
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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
