A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.

Let ABCD be a rectangle and let an equilateral triangle be described on side CD of the rectangle.

Let AB = x and BC = y. Let A be the area of the given figure.

Given, 3x + 2y = 12

Let ABCD be a rectangle and let an equilateral triangle be described on side CD of the rectangle. 

Let AB = x and BC = y. Let A be the area of the given figure.

Given, 3x + 2y = 12

x₁ < x₂

A = Area of equilateral triangle + Area of reactangle 

f(x₁) > f(x₂)              (Side of equilateral triangle = x)

x₁, x₂ ∈l

f’ (x) > 0

Differentiating both sides with respect to x, we get

x ∈ (a, b)