optimization

Question

Find the dimensions x and y of the rectangle inscribed in a circle of radius r=10 that maximizes the quantity xy^2

Answer 1

For the inscribed rectangle, let (x,y) be the coordinates of one vertex, both x & y being > 0.

The four vertices of the rectangle would be (±x,±y), in various combinations of signs.

 

We can say that x^{2 +} y² = 100 (eq. of circle)...so y² = 100 -x².

 

Consider g(x) = xy² = x(100 -x²) = 100x - x³ ... a function to be maximized.

 

Its derivative g'(x) = 100 - 3x² and its critical point occurs at x = 10/√3, where, BTW, g' < 0 so this C.P. is a maximum.

Thus the dimensions of the rectangle in question are horizontal side = 20/√3 or about  11.547.

 

Corresponding to the x root, the associated y value is the square root of [100 - 100/3]; this is about 8.165 and so the vertical side of the rectangle in question must be 16.330.