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

Explanation

Let the length of the window be `x m` and breadth of the window be `y m`.

Its total perimeter `=12 m`

`implies x+y+x+x+y=12implies y={12-3x}/2`

Area of the window `=xy+{sqrt3}/4x^2`

`implies A=6x-3/2x^2+sqrt3/4x^2`

`{dA}/{dx}=6-x(3-sqrt3/2)`

Put `{dA}/{dx}=0`

Thus `6-x(3-sqrt3/2)=0 `

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