An equilateral triangle of side $4\sqrt[4]{3}cm$ formed out of a sheet is converted into a rectangle such that there is no loss of the area of the triangle. Then the least perimeter of the rectangle (in cm) will be:

$8\sqrt{3}$

The correct answer is Option (3) → $8\sqrt{3}$

Area = $\frac{\sqrt{3}}{4}.side^2$

$=\frac{\sqrt{3}}{4}(4\sqrt[4]{3})^2$

$=\frac{\sqrt{3}}{4}.16\sqrt{3}$

$=12cm^2$

To minimize the perimeter the rectangle should be as clost to a square as possible.

Let $l=b$

$l.b=12⇒l^2=12$

$⇒l=2\sqrt{3}cm$

∴ The perimeter = $4l=8\sqrt{3}cm$