Max area - A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 16 feet.

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My anus is leaking acid.

let the radius of the circle be x ft

then the base of the rectangle is 2x ft
let the height of the rectangle be y ft

2x + 2y + pi x = 16
y = (16-2x- pix)/2

Area = pi x^2/2 + 2xy
= pi x^2/2 + 2x(16-2x-pix)/2
= pi x^2/2 + 16 - 2x - pix

d(Area)/dx = pix - 2 - pi
= 0 for a max of area.

etc.

To find the dimensions of a Norman window of maximum area, we need to maximize the area of the window while considering the given perimeter constraint.

Let's denote the width of the rectangular window as 2x and the height as y.

The perimeter of the rectangular window is 2x + 2y.

Since the total perimeter is given as 16 feet, we have the equation:

2x + 2y + πx = 16

Simplifying this equation, we get:

2x + 2x + πx = 16

Combining like terms, we have:

4x + πx = 16

Factoring out x, we get:

x(4 + π) = 16

Dividing both sides by (4 + π), we obtain:

x = 16 / (4 + π)

Now, to find the height y, we can substitute the value of x back into the equation 2x + 2y = 16:

2(16 / (4 + π)) + 2y = 16

Simplifying this equation, we have:

32 / (4 + π) + 2y = 16

Subtracting 32 / (4 + π) from both sides, we get:

2y = 16 - 32 / (4 + π)

Dividing both sides by 2, we obtain:

y = (16 - 32 / (4 + π)) / 2

Therefore, the dimensions of the Norman window that maximizes the area can be calculated as follows:
Width (2x) = 16 / (4 + π) feet
Height (y) = (16 - 32 / (4 + π)) / 2 feet

To find the dimensions of a Norman window of maximum area, we can use the concept of calculus. Let's solve this step by step:

Step 1: Identify the variables
- Let's call the width of the rectangular window as "x" (as shown in the diagram).
- Let's call the height of the rectangular window as "y" (as shown in the diagram).

Step 2: Formulate the equations based on the problem statement
- The perimeter of the Norman window is given as 16 feet. Therefore, we can write the equation as:
2y + x + πx/2 = 16

Step 3: Simplify the equation
2y + x + πx/2 = 16
Rearranging, we get:
x(1 + π/2) + 2y = 16
x(1 + π/2) = 16 - 2y
x = (16 - 2y) / (1 + π/2)

Step 4: Calculate the area of the Norman window
- The area of the window consists of the rectangle and the semicircle. Therefore, we can write the equation as:
Area = x * y + (π * x^2)/8

Step 5: Substitute the value of x in terms of y into the area equation
- Substitute the equation for x obtained in Step 3:
Area = [(16 - 2y) / (1 + π/2)] * y + (π * [(16 - 2y) / (1 + π/2)]^2)/8

Step 6: Differentiate the area equation
- Differentiate the equation obtained in Step 5 with respect to y to find the maximum area.
- Use the product rule, chain rule, and quotient rule of differentiation.

Step 7: Set the derivative equal to zero and solve for y
- Set the derivative of the area equation (obtained in Step 6) equal to zero to find the critical points.
- Solve for y.

Step 8: Substitute the obtained value of y back into the perimeter equation
- Substitute the value of y obtained in Step 7 into the perimeter equation (from Step 2) to find the corresponding value of x.

Step 9: Calculate the maximum area
- Substitute the values of x and y obtained in Step 8 into the area equation (from Step 4) to find the maximum area of the Norman window.

By following these steps, you should be able to find the dimensions of the Norman window that maximizes the area.