Max / mins, confusing.

[Dave J]

I have this study question here, that I am really stumped on.

It says.

A heritage home features a semicircular window of radius 1m. A man is commisioned to accent the window with a rectangular pane of stained glass. The glass will be attached to inside the window frame. What dimensions of the rectangular plane will provide the greatest possible area for the stained glass accent?

Im really not sure what its asking, the only information I get is the window's radius is 1m. I think it may be easier then Im trying it to be, but after a long time of looking at it, Im not sure how to approach it.

 

[galactus]

At first I thought this was one of those 'rectangular window attached to a

semicircular window' problems. But, it appears they want to know the

rectangle of max area which can be inscribed in a semicircle of radius 1.

That's what it amounts to.

If you center the semicircle at the origin, then you can divide the rectangle in two, which will have area A=2xy.

The equation of the semicircle is \(\displaystyle y=\sqrt{1-x^{2}}\)

You have:

\(\displaystyle \L\\A=2x(\sqrt{1-x^{2}})\)

Differentiate, set to 0 and solve for x.

 

[stapel]

Also posted (and answered) here.