A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 38 feet?
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y=6 x=4
To find the area of the largest possible Norman window with a perimeter of 38 feet, we need to determine the dimensions of the window that would maximize the area.
Let's break down the problem into the following steps:
1. Represent the dimensions of the window: Let's assume the width of the rectangle is 'w' feet. Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle will be 'w/2' feet.
2. Express the perimeter of the window in terms of 'w': The perimeter of the window is given as 38 feet. It consists of the sum of the perimeter of the rectangle and half of the circumference of the semicircle.
The perimeter of the rectangle is 2w, and the circumference of the semicircle is half of the circumference of a circle, which is π times the diameter, or πw. Therefore, the perimeter equation becomes:
2w + 0.5πw = 38
3. Solve for 'w': Rearrange the equation to solve for 'w'.
2w + 0.5πw = 38
Simplify by factoring out 'w':
w(2 + 0.5π) = 38
Divide both sides by (2 + 0.5π):
w = 38 / (2 + 0.5π)
4. Substitute 'w' into the area formula: The area of the Norman window is the sum of the area of the rectangle and the area of the semicircle.
The area of the rectangle is length times width, or lw.
The area of the semicircle is half the area of a circle, which is 0.5πr^2, where 'r' is the radius of the semicircle.
Substituting the values, the formula for the area becomes:
A = lw + 0.5πr^2
A = w(w/2) + 0.5π(w/2)^2
Now, we can substitute the value of 'w' from step 3 into the equation to get the final formula for the area.
A = (38 / (2 + 0.5π)) * (38 / (2 + 0.5π)) + 0.5π * (38 / (2 + 0.5π) / 2)^2
Simplifying this equation will give the area of the largest possible Norman window.