A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 16.200 ft. give the area A of the window in square feet when the width is 3.000 ft.
To find the area of the Norman window, we need to determine the dimensions of the rectangle and the semicircle.
Let's assume that the width of the rectangle is "w" ft. Since the width is given as 3.000 ft, we can substitute this value into the equation: w = 3.000 ft.
The length of the rectangle can be calculated using the perimeter formula for a rectangle: Perimeter = 2*length + 2*width. In this case, the perimeter is given as 16.200 ft, so we can substitute the known values: 16.200 ft = 2*length + 2*3.000 ft.
Simplifying the equation, we have: 16.200 ft = 2*length + 6.000 ft.
Subtracting 6.000 ft from both sides of the equation, we get: 10.200 ft = 2*length.
Dividing both sides of the equation by 2, we find the length of the rectangle: length = 5.100 ft.
Now let's determine the radius of the semicircle. The radius is equal to half the width, so the radius is 3.000 ft / 2 = 1.500 ft.
To calculate the area of the semicircle, we can use the formula: A = π * r^2 / 2, where A is the area and r is the radius. Substituting the known values: A = π * (1.500 ft)^2 / 2.
Calculating the area of the semicircle, we get: A = 3.14159 * 1.500 ft^2 / 2.
Simplifying the expression, we obtain: A ≈ 3.53429 ft^2.
Finally, we can calculate the total area of the Norman window by summing the area of the rectangle and the semicircle: A = rectangle area + semicircle area.
The rectangle area is calculated as length * width: rectangle area = 5.100 ft * 3.000 ft.
Substituting the known values, we get: rectangle area = 15.300 ft^2.
Adding the area of the rectangle and the semicircle, the total area of the Norman window is: A ≈ 15.300 ft^2 + 3.53429 ft^2.
Calculating the total area, we find that A ≈ 18.83429 ft^2.
Therefore, when the width of the Norman window is 3.000 ft, the area of the window is approximately 18.83429 square feet.
To find the area A of the Norman window, we need to break down the shape into a rectangle and a semicircle, and then calculate the total area.
1. Start by finding the dimensions of the rectangle.
Let the width of the rectangle be W = 3.000 ft.
2. The length of the rectangle can be calculated using the perimeter formula: P = 2L + 2W.
Given that the perimeter is 16.200 ft, we substitute the known values:
16.200 ft = 2L + 2(3.000 ft).
Rearranging the equation, we get:
16.200 ft - 2(3.000 ft) = 2L.
16.200 ft - 6.000 ft = 2L.
10.200 ft = 2L.
Divide both sides of the equation by 2 to isolate L:
L = 10.200 ft / 2.
L = 5.100 ft.
So, the length of the rectangle is 5.100 ft.
3. Now we find the area of the rectangle:
Area of a rectangle = Length × Width.
Area of the rectangle = 5.100 ft × 3.000 ft.
Calculating, we get:
Area of the rectangle = 15.300 sq. ft.
4. Next, we find the area of the semicircle.
The semicircle is positioned on top of the rectangle, sharing the same width.
The diameter of the semicircle is equal to the width of the rectangle, which is 3.000 ft.
The formula for the area of a semicircle is:
Area of a semicircle = (π × r^2) / 2.
Here, the radius (r) is half the length of the semicircle, which is also half the width of the rectangle.
Radius (r) = 1/2 × 3.000 ft = 1.500 ft.
Substituting the values into the formula:
Area of the semicircle = (π × (1.500 ft)^2) / 2.
Area of the semicircle ≈ (3.14 × 2.250 sq. ft) / 2.
Area of the semicircle ≈ 3.526 ft^2.
5. Finally, we calculate the total area of the Norman window by adding the area of the rectangle and the area of the semicircle:
Total area A = Area of the rectangle + Area of the semicircle.
Total area A ≈ 15.300 sq. ft + 3.526 sq. ft.
Calculating, we get:
Total area A ≈ 18.826 sq. ft.
Therefore, the area A of the Norman window, when the width is 3.000 ft, is approximately 18.826 square feet.