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The section of a window consists of a rectangle surmounted by an equilateral triangle. If the perimeters be given as $16 m$ , find the dimensions of the window in order that the maximum amount of light may be admitted.

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2, 5

3, 3.5

3.75, 2.375

4, 2

answer is C.

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Detailed Solution

Given,
Window’s perimeter

= 16 m

Suppose ABCD is a rectangle and CDE is an equilateral triangle.
Question Image

In this figure,

AB = x

BC = y

As, perimeter is 16 m. Thus,

\Rightarrow 3 x + 2 y = 16

\Rightarrow y = \frac{16 - 3 x}{2}

….(1)
Area of total figure (A) is sum of the area of triangle and area of rectangle. Thus,

\Rightarrow A = \frac{\sqrt{3}}{4} x^{2} + xy

(since area of equilateral △ = \frac{\sqrt{3}}{4} x^{2} area of reactangle = xy)

Put value of y from equation 1,

\Rightarrow A = \frac{\sqrt{3}}{4} x^{2} + (\frac{16 - 3 x}{2}) . . . . . (2)

Differentiate it with respect to x,

\Rightarrow \frac{dA}{dx} = \frac{d}{dx} (\frac{\sqrt{3}}{4} x^{2} + (\frac{16 x - 3 x^{2}}{2}))

\Rightarrow \frac{dA}{dx} = \frac{\sqrt{3}}{4} (2 x) + (\frac{16 - 16 x}{2})

\Rightarrow \frac{dA}{dx} = \frac{\sqrt{3}}{2} (x) + 8 - 3 x . . . . . . (2)

Now,

= \frac{\sqrt{3}}{2} (x) + 8 - 3 x = 0

(

\frac{dA}{dx} = 0

for maxima or minima)

\Rightarrow x (3 - \frac{\sqrt{3}}{2}) = 8 \Rightarrow x (\frac{6 - \sqrt{3}}{2}) = 8 \Rightarrow x (6 - \sqrt{3}) = 16 \Rightarrow x = \frac{16}{(6 - \sqrt{3})}

Again, differentiate (2) with respect to x,

\frac{d^{2} A}{d x^{2}} = \frac{\sqrt{3}}{2} - 3 < 0

Thus, area is maximum,

x = \frac{16}{6 - \sqrt{3}}

Put this value in equation 1,

\Rightarrow y = \frac{16 - 3 (\frac{16}{6 - \sqrt{3}})}{2}

\Rightarrow y = \frac{16 - \frac{48}{6 - \sqrt{3}}}{2} \Rightarrow y = \frac{16 (6 - \sqrt{3}) - 48}{2 (6 - \sqrt{3})}

\Rightarrow y = \frac{96 - 16 \sqrt{3} - 48}{2 (6 - \sqrt{3})}

\Rightarrow y = \frac{48 - 16 \sqrt{3}}{2 (6 - \sqrt{3})}

\Rightarrow y = \frac{24 - 8 \sqrt{3}}{(6 - \sqrt{3})}

Thus,
Breadth,

x = \frac{16}{(6 - \sqrt{3})} \approx 3.75

Length,

y = \frac{24 - 8 \sqrt{3}}{(6 - \sqrt{3})} \approx 2.375

Hence, the correct option is (3).

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