A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 50.700 ft. give the area A of the window in square feet when the width is 8.800 ft. Give the answer to two decimal places.
If the rectangular part has width x and height y, then we have a perimeter
x+2y+πx/2 = 50.700
8.8+2y+8.8π/2 = 50.7
y = 14.039
So, the area is
A = xy + π/2 (x/2)^2
= 8.8*14.039 + π/2 * 4.4^2
= 153.954
To find the area (A) of the Norman window, we need to break it down into two parts: the rectangular part and the semicircular part.
First, let's find the dimensions of the rectangle. The width of the window is given as 8.800 ft, so that will be the width of the rectangle. Since the shape is a rectangle surmounted by a semicircle, the height of the rectangle is half the height of the semicircle.
To find the height of the semicircle, we need to calculate the radius of the circle since the radius of the semicircle is equal to the width of the rectangle. So, the radius (r) is also 8.800 ft.
Now, let's find the circumference of the semicircle. The circumference of a full circle is given by the formula C = 2πr. Since we have a semicircle, we only need half of the circumference. Therefore, the circumference of the semicircle is πr.
The perimeter of the window is given as 50.700 ft, which is equal to the sum of the lengths of all its sides. So, we can set up an equation to find the length of the sides:
50.700 ft = 2 * width of the rectangle + circumference of the semicircle
50.700 ft = 2 * 8.800 ft + π * 8.800 ft
Now, let's solve this equation to find the value of π (pi). Rearranging the equation, we have:
50.700 ft = 17.600 ft + π * 8.800 ft
33.100 ft = π * 8.800 ft
Dividing both sides by 8.800 ft, we get:
π = 33.100 ft / 8.800 ft
π ≈ 3.75
Now, let's substitute the value of π back into our original equation to find the length of the sides:
50.700 ft = 2 * 8.800 ft + 3.75 * 8.800 ft
50.700 ft = 17.600 ft + 32.7 ft
50.700 ft = 50.300 ft + 32.7 ft
50.700 ft = 50.300 ft + 21.0 ft + 11.7 ft
So, the length of the sides of the window are approximately 21.0 ft and 11.7 ft.
Now that we have the dimensions of the rectangle (8.800 ft x 21.0 ft) and the radius of the semicircle (8.800 ft), we can calculate the area of the window.
The area of the rectangle is given by the formula A_rectangle = length * width.
A_rectangle = 8.800 ft * 21.0 ft
The area of the semicircle is given by the formula A_semicircle = 1/2 * π * r^2.
A_semicircle = 1/2 * 3.75 * (8.800 ft)^2
Finally, we add the area of the rectangle and the area of the semicircle to find the total area of the Norman window:
A = A_rectangle + A_semicircle
Plug in the values we found and calculate the area A to two decimal places.