Mathematics: Anorman window has the shape of a rectangle surmounted by a semicircle. (thus the diameter of the semicircle is equal to the width of the rectangle.) if the perimeter of the window is 16 ft, find the value of x so that the greatest possible amount of light is admitted. (give your answer correct to two decimal places.) x = ft

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Anorman window has the shape of a rectangle surmounted

Ответ:

dawnmyers88

11.03.2022

Mathematics

Answer-

For x = 2.24, greatest possible amount of light will be admitted.

Solution-

According to the question, the perimeter of the window is 16 ft

From the attachment,

Perimeter of the window = Perimeter of the rectangle-Width of the rectangle + Curved perimeter of the semi-circle

So,

$\Rightarrow (2x+y+y+2x)-(2x)+(\pi x)=16\\\\\Rightarrow 2x+2y+\pi x=16\\\\\Rightarrow 2y=16-2x-\pi x\\\\\Rightarrow y=\dfrac{16-2x-\pi x}{2}\\\\\Rightarrow y=8-x-\dfrac{\pi x}{2}$

As we have to find the value of x so that the greatest possible amount of light is admitted, so we have to calculate the maximum area for which the perimeter is 16 ft.

$\text{Area of the window}=\text{Area of the rectangle + Area of the semi-circle}$