document.write( "Question 785527: The perimeter of a triangle is 75 feet. What is the largest possible area? \n" ); document.write( "

Algebra.Com's Answer #477681 by KMST(5328) $\"\"$ $\"About$

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The largest possible area requires an equilateral triangle.
\n" ); document.write( "An equilateral triangle with a perimeter of 75 feet eill have side length of
\n" ); document.write( " $\"75%2F3\"$ $\"feet=25feet\"$
\n" ); document.write( "The area of a triangle with sides of length $\"a\"$ and $\"b\"$ forming an angle $\"C\"$ is
\n" ); document.write( " $\"%281%2F2%29%2Aa%2Ab%2Asin%28C%29\"$
\n" ); document.write( "The angles in an equilateral triangle measure $\"60%5Eo\"$
\n" ); document.write( "The area of our largest triangle is
\n" ); document.write( "

$\"square+feet\"$
\n" ); document.write( " $\"625sqrt%283%29%2F4=156.25sqrt%283%29\"$ would be the exact number of square feet, but I would round/approximate it as 271 square feet.
\n" ); document.write( "
\n" ); document.write( "How do I know we need an equilateral triangle?
\n" ); document.write( "Heron's formula says that the area of a triangle with side length a, b, and c is
\n" ); document.write( " $\"sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29\"$ , where $\"s=%28a%2Bb%2Bc%29%2F2\"$ is the semi=perimeter (half of the perimeter).
\n" ); document.write( "If the perimeter is fixed (75 feet), s = 37.5 feet, is also fixed.
\n" ); document.write( "For maximum area we need to maximize $\"%28s-a%29%28s-b%29%28s-c%29\"$
\n" ); document.write( "I am not sure how to prove it with 3 dimensions, but since enlarging one of those factors requires making one or both of the other two smaller, I believe that the maximum product requires s-a=s-b=s-c, which translates into a=b=c. \n" ); document.write( "

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