Compute the sides and volume of the largest prism that can be $t$ between a plane and a hemisphere of radius $2^{1/2}$ on the plane

Question

Compute the sides and volume of the largest prism that can be $t$ between a plane and a hemisphere of radius $2^{1/2}$ on the plane.

Hint: Argue that $$A(x; y) = xy(2-x^2-y^2)$$ is the function which describes the volume of the wanted prism.