The sum of the perimeters of an equilateral triangle and a square is 24. Find the dimensions of the triangle and the square that produce a minimum total area.

Question

The sum of the perimeters of an equilateral triangle and a square is 24. Find the dimensions of the triangle and the square that produce a minimum total area.

Answer 1

Let x = the side length of the equilateral triangle; then its area is ¼x²√3 and the square's perimeter is 24-3x.

Therefore the side of the square is 6-¾•x and therefore the area of the square is (6-½•x)².

 

You desire to minimize the total area function f(x) = ¼x²√3 + (6-¾•x)².

 

Use your knowledge of calculus, now, beginning by setting f' to zero, etc.