Question 1031407: Complete the following steps in order to determine whether c feet of fencing will enclose more area when arranged as a square or as a circle.
A.) Express the length of a side of a square (s) and of the radius of a circle (r) in terms of c.
B.) Express the area of the square and of the circle in terms of x
C.) Which area is larger? Justify your answer
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the fencing is c feet long.
the circumference of a circle is equal to 2 * pi * r.
you get c = 2 * pi * r.
solve for r to get r = c / (2 * pi).
the perimeters of a square is equal to 4 * s.
you get c = 4 * s.
solve for s to get s = c / 4.
the area of a circle is equal to pi * r^2.
if you let x equal the area of the circle, then x = pi * r^2.
since r = (c / (2 * pi)), then x = pi * (c / (2 * pi))^2.
simplify this to get x = (pi * c^2) / (4 * pi^2).
simplify further to get x = c^2 / (4 * pi).
this means the area of the circle is equal to c^2 / (4 * pi).
the area of a square is equal to s^2.
if you let x equal the area of the square, then x = s^2.
since s = c / 4, then x = (c / 4)^2.
simplify this to get x = c^2 / 16.
this means the area of the square is equal to c^2 / 16.
the area of the square is equal to c^2 / 16.
the area of the circle is equal to c^2 / (4 * pi).
since 16 is greater than (4 * pi = 12.56), then the area of the circle, with the same perimeter as the square, is larger.
this is because the same numerator with a smaller denominator is larger.
to confirm, use some numbers.
assume the length of the fence is 60 feet.
if this fence surrounds the circle, then it becomes the circumference of the circle.
c = circumference of the circle.
c = 2 * pi * r becomes 60 = 2 * pi * r.
solve for r to get 60 / (2 * pi) = 9.549.
the area of the circle is equal to pi * r^2 = 286.479 square feet.
p = perimeter of the square.
p = 4 * s
solve for s to get s = p / 4 = 60 / 4 = 15
area of the square is equal to s^2 = 15^2 = 225 square feet.
the area of the circle is larger.
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