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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.

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Best answer

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 \(y=\frac{18-6\sqrt{3}}{6-\sqrt{3}}.\)

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