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 22 feet?

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 possibe Norman window with a perimeter of 21 feet?

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To find the area of the largest possible Norman window with a perimeter of 22 feet, we can use calculus. Let's break down the problem step by step:

1. Start by drawing a picture of the Norman window. We have a semicircle atop a rectangle, with the diameter of the semicircle equal to the width of the rectangle.

2. Let's denote the width of the rectangle as "w". Since the diameter of the semicircle is equal to the width of the rectangle, the radius of the semicircle is "w/2".

3. The perimeter of the Norman window consists of the circumference of the semicircle and the sum of the lengths of all four sides of the rectangle. Using this information, we can write the equation for the perimeter as:

Perimeter = (1/2)π(w/2) + 2w + (1/2)π(w/2) + 2w

4. Simplify the equation for the perimeter:

Perimeter = π(w/4) + 4w

5. Set up an equation to find the value of "w" that maximizes the area:

Area = (1/2)π(w/2)^2 + w^2

6. Simplify the equation for the area:

Area = (1/8)πw^2 + w^2

7. To find the maximum area, we need to find the value of "w" that maximizes the area. To do this, we can take the derivative of the area equation with respect to "w" and set it equal to zero:

d(Area)/dw = (1/8)π(2w) + 2w = 0

8. Solve the equation to find the critical point:

(1/4)πw + 2w = 0
(1/4)πw = -2w
w = -(8/π)

9. Since the width of the rectangle cannot be negative, we disregard the negative value.

10. The width of the rectangle is w = 8/π.

11. Substitute the value of "w" into the equation for the perimeter to find the actual perimeter of the Norman window:

Perimeter = π((8/π)/4) + 4(8/π) = 2 + 32/π

12. Calculate the total area using the width and the previously-determined value of "w":

Area = (1/8)π(w^2) + w^2
Area = (1/8)π((8/π)^2) + (8/π)^2

13. Simplify the equation and calculate the area:

Area = (1/8)(64/π) + (64/π)
Area = (64/8π) + (64/π)
Area = (8 + 64/π) square feet

Therefore, the area of the largest possible Norman window with a perimeter of 22 feet is (8 + 64/π) square feet.

Write equations for:

(1) area of the window in terms of the semicircle diameter (x) and height (y) of the rectangle.

(2) Perimeter = constant = 22 ft

A = (pi/2)*x^2 + x*y= (pi +1)x + 2y
P = (pi+1)x + 2y = 22

Eliminate the variable y by making the substitution
y = 11 - [(pi+1)/2]x into the first equation for (A), making it an equation in one variable, x.
Solve dA/dx = 0 for the optimum x.

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