Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of $t$ .
A) $V=\frac{1}{3} \pi t^{3}$
B) $V=\frac{1}{4} \pi t^{3}$
C) $V=\frac{1}{3} \pi t^{2}$
D) $V=\frac{1}{6} \pi t^{2}$
E) $V=\frac{1}{6} \pi t^{3}$

Question

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h as shown in the figure. Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of $t$ . 
A) $V=\frac{1}{3} \pi t^{3}$ 
B) $V=\frac{1}{4} \pi t^{3}$ 
C) $V=\frac{1}{3} \pi t^{2}$ 
D) $V=\frac{1}{6} \pi t^{2}$ 
E) $V=\frac{1}{6} \pi t^{3}$

Quizplus · Accepted Answer

The volume of a cylindrical shell is given by $V=2\pi rh\cdot t$. The radius of the spherical shell is $D-t$ and the height of the cylindrical shell is $h=t$. Thus, the volume of the napkin ring is given by
$$V=2\pi(t)(D-t)\cdot t=\boxed{\frac16\pi t^3}$$
which is choice E.

Suppose You Make Napkin Rings by Drilling Holes with Different ttt

Suppose You Make Napkin Rings by Drilling Holes with Different $t$