SOLUTION: The perimeter of a triangle is 75 feet. What is the largest possible area?

Question 785527: The perimeter of a triangle is 75 feet. What is the largest possible area?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
The largest possible area requires an equilateral triangle.
An equilateral triangle with a perimeter of 75 feet eill have side length of

The area of a triangle with sides of length and forming an angle is

The angles in an equilateral triangle measure
The area of our largest triangle is

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
, where 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
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.
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