# Optimization (Math)

posted by .

The mayor of a village wants to build a library of which the windows have a shape of a rectangle on top of a square. The total perimeter of each window is of P meters and varies depending on each windows size. Find the dimensions of the windows in terms of P, that maximizes the area of all the windows.

Because I can't draw on here, the perimeter I have is

6x + 2y = P

(3x for each of the squares sides, because the rectangle covers the last one, 2y for each of the rectangles sides, and the last 3x are the rectangles length, x being half the length of the rectangle, so 2x on top of the rectangle and 1x at the bottom, because the other x is the junction between the square and the rectangle.)

-----------|
| |-------|
|___|

For the area, I found:

p = 6x +2y
y = (P-6x)/2

so

A(x) = x^2 + 2x((P-6x)/2)

now, I know I have to derive A(x)
but what I'm not sure is if the derivative of P will be 0, or do I simply leave P as it is?

• Optimization (Math) -

sorry for the drawing, apparently it didn't work when I submitted the question!

• Optimization (Math) -

A(x) looks good.
A(x) = x(P-5x) = Px - 5x^2

You can take the derivative if you want, getting

A' = P - 10x
so, A' = 0 when x = P/10

Or, you can see that the roots of A(x) are 0 and P/5, making the max halfway between, or at x = P/10
y = (P-6x)/2 = (P-P/10)/2 = 9P/20

• Optimization (Math) -

Rats. y = (P-6P/10)/2 = 4P/20 = P/5
But then, we knew that, since y=2x

• Optimization (Math) -

I understand now, thanks a lot steve!

## Similar Questions

If you have a rectangle with 24 square units 3 lines down 8 across how can you have a 4 unit rectangle with a perimeter of 8 around the outside also 6 units with a perimeter of 10 in a seprate problem A square of 4, perimeter of 8. …
2. ### Calculus

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30ft, find the dimensions of the window so that …
3. ### calculus

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30ft, find the dimensions of the window so that the greatest possible amount of light is admitted. I keep screwing up in creating …
4. ### Algebra

Window World, Inc. is responsible for designing windows for the expansion of the campus chapel. The current design is shown in the figure. The metal trim used to secure the perimeter of the frame is 126 inches long. If the maximum …
5. ### Calculus

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 25 ft, find the dimensions of the window so that …
6. ### math

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.
7. ### MATH HELP

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 50.700 ft. give the area A of the window in square feet when the width is 8.800 ft. Give the answer to two decimal places.
8. ### Calculus - Optimization

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?
9. ### Calculus

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 20 ft, find the dimensions of the window so that …
10. ### Calculus

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 20 ft, find the dimensions of the window so that …

More Similar Questions