Let x and y be the dimensions of rectangular part of window and x be side of equilateral part.
If A be the total area of window, then
\(A=x.y+\frac{\sqrt3}{4}x^2\) ....(i)
Also,
x + 2y + 2x = 12
⇒ 3x +2y = 12
[Differentiating with respect to x]
Now, for maxima or minima
Again,
i.e., maximum if \(x=\frac{12}{6-\sqrt{3}}\) and
i.e., For largest area of window, dimensions of rectangle are
\(x=\frac{12}{6-\sqrt{3}}\) and \(y=\frac{18-6\sqrt{3}}{6-\sqrt{3}}.\)