optimization: rectangle within a semi-circle

[Vempy]

Question:
A heritage home features a semicircular window of radius 1m. A local artisan is comissioned to accent the window with a rectangular pane of stained glass. The stained glass will be attached to the inside of the window frame. What dimensions of the rectable will provide the greatest possible area for the stained glass accent?

So the bottom of the semi-circle is 2m in total (2r), and thus the rectangle's length will be 2-2x, or 1-x.

The height of the window is 1m, and thus the height of the rectangle is 1-y.

Area of a semi-circle is:
A = 1/2pir^2
A = 1/2pi1^1
A = 1/2pi or 1.57m^2

Area of rectangle:
A = bh
A = (1-x)(1-y)

But, I'm pretty sure that isn't right in some way.

 

[soroban]

Hello, Vempy!

Your reasoning is way off . . .

A heritage home features a semicircular window of radius 1m.
A local artisan is comissioned to accent the window with a rectangular pane of stained glass.
The stained glass will be attached to the inside of the window frame.
What dimensions of the rectangle will have the greatest possible area for the stained glass?

                |
              * * *
          *-----+-----*
        * |     |     | *
       *  |     |    y|  *
          |     |     |
      *   |     |     |   *
    - * - + - - + - - + - * -
             x  |  x

The length of the rectangle is \(\displaystyle 2x\); its height is \(\displaystyle y.\)

The equation of the circle is: \(\displaystyle \,x^2\,+\,y^2\:=\:1\;\;\Rightarrow\;\;y \:=\:\sqrt{1\,-\,x^2}\;\) [1]

The area of the rectangle is: \(\displaystyle \,A\;=\;2xy\;\) [2]


Substitute [1] into [2]: \(\displaystyle \:A \:=\:2x\left(1\,-\,x^2\right)^{\frac{1}{2}}\)

And that is the function we must maximize . . .

 

[Vempy]

Hey!

Wow, thanks. I was kind of off, eh?

Thank you a bunch!