Rectangle is inscribed in a equilateral triangle.

$\"\"$

$\"\"$ and $\"\"$ are the side of the rectangle.

$\"\"$ be the one of the side of the equilateral triangle.

Observe the figure.

$\"\"$

The area of the rectangle is $\"\"$ .

$\"\"$

The area of the rectangle is maximum when its first derivative becomes zero.

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

Equate $\"\"$ to zero.

$\"\"$

The length of the rectangle is $\"\"$ .

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

$\"\"$

The width of the rectanngle is $\"\"$ .

The length of the rectangle is $\"\"$ .

The width of the rectanngle is $\"\"$ .