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

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 $t$ 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 $t$ .


Select the correct answer.

A) $V = \frac { 1 } { 6 } \pi t ^ { 2 }$
B) $V = \frac { 1 } { 3 } \pi t ^ { 2 }$
C) $V = \frac { 1 } { 6 } \pi t ^ { 3 }$
D) $V = \frac { 1 } { 3 } \pi t ^ { 3 }$
E) $V = \frac { 1 } { 4 } \pi t ^ { 3 }$