$\"\"$

\

The sum of perimeters of an equilateral traingle and a square is $\"\"$ .

\

Area of the square is $\"\"$ .

\

Area of the equilateral traingle is $\"\"$ .

\

Let us consider side of the equilateral triangle that produces minimum area is $\"\"$ .

\

Let us consider side of the square that produces minimum area is $\"\"$ .

\

Total area : $\"\"$ .

\

Sum of perimeters of an equilateral traingle and a square is $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Find the minimum area.

\

$\"\"$

\

Differentiate on each side with respect to $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Evaluate $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

The dimensions of the triangle minimum area is $\"\"$ .

\

$\"\"$

\

Substitute $\"\"$ in $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The dimensions of the square minimum area is $\"\"$ .

\

$\"\"$

\

The dimensions of the triangle for minimum area is $\"\"$ .

\

The dimensions of the square for minimum area is $\"\"$ .