Archimedes' Method for Computing Areas and Volumes - Exercise on Proposition 4 of The Method

Investigate Archimedes' Proposition 4 below, which concerns a paraboloid (the solid obtained by rotating the curve \(x=y^2\) about the \(x\)-axis) inscribed inside a cylinder (with radius \(CD\) and height \(AD\)). 

a) Describe the balancing relationship that is illustrated in the sketch.

b) Use this relationship to determine a relationship between the volume of the paraboloid and the volume of the cylinder.

c) Find a formula for the volume of the paraboloid in terms of \(r\), the radius of the cylinder.

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