Word Area Problem

Question

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 840 centimeters and the length of the base of the window is 122 centimeters, find the total area of the window.

Answer 1

we have the length of the base which is 122 cm. We double this and subtract from 840 cm to get the sum of the length of the sides, so:

 

840 - 2(122) = 596

 

to get length of one side we divide 596 by 2, so:

 

596/2 = 298

 

Now, we have the length of side and base which are: 298 cm and 122 cm.

 

To get area of the window we multiply the lengths: 298 * 122 = 36,356 cm²

 

To get the area of the semicircles, we need the radius of the semicircles which is 122/2 = 61 cm.

 

The area of a semicircle is ∏r²/2 = ∏(61²)/2 ------>1860.5∏

 

So, total area is (36,356 + 1860.5∏) cm²

 

 

Answer 2

Here's what we need to find out:

A = l * w + (Pi * r²)/2 _{(the area of a rectangle with half the area of a circle)}

So we need to find out what w is

 

P = l + 2w + pi * r _{(but not a second length because the semicircle is covering it, hence the Pi * r in its place)}

here's what we know already:

l = 122

r =  l / 2 = 61

P = 840

So we have:

840 = 122 + 2w + 3.14 * 61

840 = 122 + 2w + 191.54

840 = 313.54 + 2w

526.46 = 2w

263.23 = w

 

Now that we know w, we can solve the area:

A = l * w + (Pi * r²)/2

A = 122 * 263.23 + (3.14 * 61²)/2

A = 32,114.06 + 11683.94 / 2

A = 32114.06 + 5841.97

A = 37,956.03 cm²