A cube of maximum possible volume is cut from the solid right circular cylinder. What is the ratio of volume of cube to that of cylinder if the edge of a cube is equal to the height of the cylinder?

In a right angle triangle $\mathrm{ABC}$, what is the maximum possible area of a square that can be inscribed when one of its vertices coincide with the vertex of the right angle of the triangle?

In the adjoining figure, a square of maximum possible area is circumscribed by the right-angle triangle $\mathrm{ABC}$ in such a way that one of its sides just lies on the hypotenuse of the triangle. What is the area of the square?

In the adjoining figure P Q R S is a square of maximum possible area which is circumscribed by the semicircle. Points R and S lie on the diameter A  B. What is the area of the square if the radius of the circle is ' r ?

In the adjoining figure a quadrant (of circle) inscribes a square of maximum possible area. If the radius of the circle be ' $r$ ' then what is the area of the square?

In the adjoining figure, $\mathrm{AB}$ is the diameter of a semicircle which inscribes a circle of maximum possible area. If the radius of the larger circle (i.e., semicircle) is $\mathrm{r}$, the area of the inscribed circle is:

In a quadrant (of a circle) a circle of maximum possible area is given. If the radius of the circumscribing quadrant be r, then what is the area of the inscribed circle?