Question 785527
The largest possible area requires an equilateral triangle.
An equilateral triangle with a perimeter of 75 feet eill have side length of
{{{75/3}}}{{{feet=25feet}}}
The area of a triangle with sides of length {{{a}}} and {{{b}}} forming an angle {{{C}}} is
{{{(1/2)*a*b*sin(C)}}}
The angles in an equilateral triangle measure {{{60^o}}}
The area of our largest triangle is
{{{(1/2)*(25feet)*(25feet)*sin(60^o)=(1/2)*(625)*(sqrt(3)/2)}}}{{{square feet}}}
{{{625sqrt(3)/4=156.25sqrt(3)}}} would be the exact number of square feet, but I would round/approximate it as 271 square feet.
 
How do I know we need an equilateral triangle?
Heron's formula says that the area of a triangle with side length a, b, and c is
{{{sqrt(s(s-a)(s-b)(s-c))}}}, where {{{s=(a+b+c)/2}}} is the semi=perimeter (half of the perimeter).
If the perimeter is fixed (75 feet), s = 37.5 feet, is also fixed.
For maximum area we need to maximize {{{(s-a)(s-b)(s-c)}}}
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.