A Norman window has the shape of a square with a semicircle
mounted on it. Find the width of the window if the total area of
the square and the semicircle is to be 240 ft squared
To find the width of the window, we need to determine the dimensions of both the square and the semicircle.
Let's denote the width of the square as x. Since the window is in the shape of a square, the length would also be x.
The area of the square is given by A_square = x * x = x^2.
Now, let's consider the semicircle. The width of the semicircle is the same as the width of the square, which is x. Therefore, the radius of the semicircle would be half of the width, which is x/2.
The area of the semicircle can be calculated using the formula A_semicircle = (π * r^2) / 2, where r is the radius.
Plugging in the values, we get A_semicircle = (π * (x/2)^2) / 2 = (π * (x^2)/4) / 2 = (π * x^2) / 8.
The total area of the square and the semicircle is given as 240 ft^2, so we can set up the equation:
x^2 + (π * x^2) / 8 = 240.
To solve this equation, we need to find the value of x that satisfies this equation.
Multiplying both sides of the equation by 8 to eliminate the fraction:
8x^2 + πx^2 = 1920.
Combining the like terms:
x^2 (8 + π) = 1920.
Dividing both sides by (8 + π):
x^2 = 1920 / (8 + π).
Taking the square root of both sides to isolate x:
x = √(1920 / (8 + π)).
Using a calculator, the value of x is approximately 12.65 ft (rounded to two decimal places).
Therefore, the width of the window is 12.65 ft.
To find the width of the Norman window, we need to use the given total area of 240 ft².
Let's break down the window into its components: a square and a semicircle.
1. Square:
The area of a square is given by A = s^2, where s is the side length.
Let the side length of the square be x. Therefore, the area of the square is x^2.
2. Semicircle:
The area of a semicircle is half the area of a full circle with the same radius.
The formula for the area of a circle is A = πr^2, where r is the radius.
In our case, the radius of the semicircle is equal to half the side length of the square, so the radius is (x/2).
Thus, the area of the semicircle is (1/2) * π * (x/2)^2.
Now, we can add the areas of the square and the semicircle to get the given total area:
x^2 + (1/2) * π * (x/2)^2 = 240
Simplifying the equation further:
x^2 + (π/8) * x^2 = 240
Combining the terms with x^2:
(1 + π/8) * x^2 = 240
Now, let's solve for x by isolating it:
x^2 = 240 / (1 + π/8)
x = √(240 / (1 + π/8))
x ≈ 10.35 ft (rounded to two decimal places)
Therefore, the width of the window, which is the side length of the square, is approximately 10.35 feet.
w^2 + (π/2) (w/2)^2 = 240 ft^2
w^2 [1 + (π/8)] = 240