Question 181013: If I have 100 ft. of fencing and I want to enclose the most area, should I make the enclosure a circle, triangle, or a square Prove your answer.
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! If I have 100 ft. of fencing and I want to enclose the most area, should I make the enclosure a circle, triangle, or a square Prove your answer.
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With only 3 to choose from, it's a matter of determining area vs. perimiter.
Circle:
C = 2*pi*r = 100
r = 50/pi
Area = pi*r^2
Area = pi*(50/pi)^2
Area = 2500/pi =~795.8 sq ft
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Square:
C = 4s = 100 feet
s = 25 ft
Area = 25^2 = 625 sq ft
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Triangle (equilateral):
C = 3s = 100 feet
s = 100/3 feet
Area = bh/2
Area = s*(s/2)*sqrt(3)/2
Area = s^2*sqrt(3)/4
Area =~ 481 sq ft
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The circle encloses the greatest area.
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The enclosed area increases with the number of sides of the polygon, and is a maximum when the # of sides is infinite, which is a circle.
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