A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 58.9 ft. give the area A of the window in square feet when the width is 12 ft. Give the answer to two decimal places.
let the radius of the semicircle be x
then the width of the rectangle is 2x
let the height of the rectangle be y
(1/2)(2π(x)) + 2x + 2y = 58.9
πx + 2x + 2y = 58.9
when y = 12,
x(π+2) + 24 = 58.9
x(π+2) = 34.9
x = 6.79
area of window = (1/2)π(6.79)^2 + 12(6.79) = .....
when x = 12
y = 58.9 - 24 - 12π = appr
To find the area (A) of a Norman window, we need to separate it into two parts: the rectangle and the semicircle.
Step 1: Find the length of the window.
The perimeter of the window is the sum of the lengths of all its sides. Given the perimeter is 58.9 ft, we can form an equation:
2 * width + semicircle circumference = 58.9
Since the width is given as 12 ft, we substitute the value in the equation:
2 * 12 + semicircle circumference = 58.9
24 + semicircle circumference = 58.9
semicircle circumference = 58.9 - 24
semicircle circumference = 34.9 ft
Step 2: Find the radius of the semicircle.
The circumference of a semicircle is π * diameter. In this case, the circumference is 34.9 ft. We can write an equation:
π * diameter = 34.9
To find the radius, divide the diameter by 2:
radius = diameter / 2 = 34.9 / (2 * π)
Step 3: Calculate the area of the rectangle.
The area of a rectangle is length × width. The width is given as 12 ft, and we need to find the length. As the length is the same as the width, the area of the rectangle is:
rectangle area = length × width = 12 ft × 12 ft
Step 4: Calculate the area of the semicircle.
The area of a semicircle is 1/2 × π × radius^2. We have already calculated the radius in Step 2. Plug the value into the formula:
semicircle area = 1/2 × π × radius^2
Step 5: Calculate the total area of the window.
To find the total area of the window, add the area of the rectangle and the area of the semicircle:
window area = rectangle area + semicircle area
Now, we can plug in the values and calculate the area:
rectangle area = 12 ft × 12 ft
semicircle area = 1/2 × π × radius^2
window area = rectangle area + semicircle area
Plug in the values and calculate the expression to find the area of the window (A) in square feet.