Optimization:semicircular window..greatest area for accent

[wind]

Hi, I am having trouble with this question

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

so the stained glass will fit around the semicurcular window right?

the width of the rectangular pane would be the same as the diameter of the circle
the lenght of the rectangular pane would be the same as the radious of the circle

A=(r)(2r)
A=2r²

the total area would be the area of the pain subtract the area of the semicircle

A=2r² - ½pir²
A'=4r + pir
0=4r + pir

...help!

Thanks

 

[tkhunny]

I think you're backwards. The rectangle fits inside the semicircle.

Give it another go.

 

[wind]

oh

Thanks tkhunny

Equation of circle at origin

x²+y²=r²
y=root (r²-x²)

Rectangle

L= y
w= 2x

A= 2xy

Area

A=2x(root r²-x²)
A=2x(r²-x²)^½

Derivitive
A'=(2)(r²-x²)^½ + (2x)[½(r²-x²)^-½ • 2r-2x]
A'=[(2)(r²-x²)^½ + (2x)(2r-2x)]/ 2(r²-x²)^-½
A'=[(2)(r²-x²)^½ + (4xr-4x²)]/ 2(r²-x²)^-½

Possible max/min
0=[(2)(r²-x²)^½ + (4xr-4x²)]/ 2(r²-x²)^-½
0=(2)(r²-x²)^½ + (4xr-4x²)

How do I evaluate this?

 

[tkhunny]

r = 1 might be a good start.

 

[TchrWill]

Re: Optimization:semicircular window..greatest area for acce

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

The maximum area rectangular figure inscribable within a circle is a square.

Since the circle has r = 1m, the square has sides of 2sqrt(2)m.

Therefore, the pane of maximum area within the semicircle is one half of this square or sqrt(2) by 2sqrt(2) or 4 sq.met.