Optimizing Rectangle Dimensions within a Circle

[rachael]

1. Find the dimensions of the rectangle with largest area which can be cut from a circle with equation x^2+ y^2= 4

this is the question but i got stuck half way when i was differentiating the equation

how do i work this out :
[square root of (4-y^2)] + ([-y ^2] \ square root[ 4- y^2])

 

[pizzasky]

So if A represents the area of the rectangle, you have \frac{dA}{dy}=4(\frac{-y^2}{\sqrt{4-y^2}}+\sqrt{4-y^2}).

If you observe the expression, is there something you can factorize that will make it easier to solve for y when you set \frac{dA}{dy}=0?

 

[arunbg]

Actually, from symmetry you can argue that the required rectangle has to be a square (special case of a rectangle) whose diagonals meet at the centre of the circle of radius 2 units .
What can you say about the length of the side of this square ?
Hint:Draw radii to the corners of the square .

Of course if the symmetry isn't apparent, you can always go for the calculus approach, which involves setting up coordinate axes and maximising .