# Math Help - greatest area inscribed in a circle

1. ## greatest area inscribed in a circle

find the rectangle of greatest area that can be inscribed in the circle x^2+y^2=36

2. The diagonal of the rectangle will be the diameter and I believe the shape is a square

3. We can always set up a coordinate system so that the origin is at the center of the circle and the x-axis is parallel to one side of the rectangle. In that coordinate system, the circle has equation $x^2+ y^2= R^2$ for some R (the radius of the circle). If we call the corner of the square in the first quadrant $(x_0, y_0)$, then, because the vertex lies oh the circle, $x_0^2+ y_0^2= R^2$ so $y_0= \sqrt{R^2- x_0^2$.

The lengths of the sides of the rectangle are, by symmetry, just twice those x and y values so the area of the rectangle is $(2x_0)(2y_0)= 4x_0\sqrt{R^2- x_0^2$. Replace $x_0$ by x and find the value of x that maximizes $A= 4x(R^2- x^2)^{1/2}$.