Exercises: Informal Arguments for Sphere Volume Using Cavalieri's Principle

Two solids each have height 8 cm. At every height $h$ between 0 and 8 cm, the cross-sectional area of Solid A equals the cross-sectional area of Solid B. By Cavalieri's principle, if Solid A has volume $320\pi$ cm³, what is the volume of Solid B in cm³?

A hemisphere of radius $r = 6$ cm is compared to a cylinder-minus-cone solid, where both the cylinder and the cone have radius 6 cm and height 6 cm. What is the volume of the cylinder-minus-cone solid in terms of $\pi$?

A hemisphere of radius $r = 5$ cm is sliced horizontally at height $h = 3$ cm above its flat base. Using the Pythagorean Theorem, find the cross-sectional area at that height. Express your answer in terms of $\pi$.

For a hemisphere of radius $r$, at height $h$ above the flat base, the cross-sectional radius $x$ satisfies   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . So the cross-sectional area is $A_{\text{hemisphere}}(h) =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

For the cylinder-minus-cone comparison solid (cylinder radius $r$, height $r$; cone apex at base, base radius $r$ at top), at height $h$ the cone's inner radius equals   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   . The annular cross-sectional area is $A_{\text{annulus}}(h) =$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Using the hemisphere argument: a hemisphere of radius $r$ has the same volume as a cylinder-minus-cone (both with radius $r$ and height $r$). Compute the hemisphere volume when $r = 3$ cm. Express your answer in terms of $\pi$.