Question

find the dimensions x and y of the rectangle inscribed in a circle of radius r that maximizes the quantity xy^2.

Answer 1

I'm not sure if this is the best way to go about this, but converting to polar coordinates lets you reason about the problem with reference to only a single variable (assuming a fixed radius r).

Answer 2

Let x = r*cos(t) and y = r*sin(t) where t represents the angle between the vector (x,y) and the positive x-axis. You can use this to get dimensions for a rectangle inscribed in a circle of radius r centered at the origin. The width of the rectangle will be w = 2x and the height h = 2y. Then you are trying to maximize wh^2.

Answer 3

Then you want to plug everything in and get to get a function f(t) = (2r*cos(t)) * (2r*sin(t))^2. To get the maximum, you can analyze the derivative of this function f'(t) = -8r^3cos(t) + 24r^3*sin(t)*cos^2(t)... I think that's right anyway, quick math. Find the critical values, i.e. angles t such that f(t) = 0. Once you found the value t that maximizes this function, you can get the maximal width and height via w=2*r*cos(t) and h=2*r*sin(t).