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nika2105 [10]
2 years ago
5

a norman window has the shape of a rectangle surmounted by a semicircle. if the perimeter of the window is 25 feet, express the

area of the window as a function of the width (across the base) of the window.
Mathematics
1 answer:
sp2606 [1]2 years ago
7 0

The area of the window is \frac{25x}{2} -(1 +\frac{\pi}{2} )\frac{x^2}{2}

The shape of the Norman window is described as a rectangle surmounted by a semicircle.

Let 'x' be the width of the window.

Then this width corresponds to the diameter of the semicircle.

That is, radius of the semicircle = x/2

Now given the perimeter of the whole window = 25 ft

This perimeter is equal to the sum of 3 sides of the rectangle and the perimeter of the semicircle.

Perimeter of the semicircle = \frac{2\pi r}{2} = \frac{\pi x}{2} , where r is the radius of the semicircle.

Now let 'y' be the length of the window.

Then sum of three sides of the window = 2y + x

So 2y + x + \frac{\pi x}{2} = 25

Therefore, y =( 25 - x - \frac{\pi x}{2} ) / 2

Area of the window = Area of the semicircle + Area of the rectangle =

 \frac{\pi x^2}{8} +xy = \frac{\pi x^2}{8} + x( 25 - x - \frac{\pi x}{2} ) / 2\\ = \frac{\pi x^2}{8} + (25x - x^2 - \frac{\pi x^2}{2} ) / 2

 So Area of the window = \frac{25x}{2} -(1 +\frac{\pi}{2} )\frac{x^2}{2}

Learn more about area of circle and rectangle at brainly.com/question/4694841

#SPJ4

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