3D figures (Cavalieri's principle) Knowledge-Based Practice

Questions

1. 1.

   Two solids of the same height lie between two parallel planes. If every plane parallel to these planes intersects the first solid in a circle of radius $3\text{ cm}$ and the second solid in a square of side length $s$, what must be the value of $s$ for the two solids to have equal volume by Cavalieri's principle?

   • A) $3\pi\text{ cm}$
   • B) $3\sqrt{\pi}\text{ cm}$
   • C) $9\pi\text{ cm}$
   • D) $6\sqrt{\pi}\text{ cm}$
2. 2.

   A cone and a right pyramid have the same height of 15 cm. If every plane parallel to their bases intersects both solids in cross sections of equal area, and the base of the pyramid has an area of $48\text{ cm}^2$, what is the volume of the cone?

   • A) $240\text{ cm}^3$
   • B) $720\text{ cm}^3$
   • C) $240\pi\text{ cm}^3$
   • D) $120\text{ cm}^3$
3. 3.

   A hemisphere of radius 6 cm and a right cylinder of radius 6 cm and height 6 cm with an inverted cone of the same base and height removed from it have equal cross-sectional areas at every level parallel to their bases. According to Cavalieri's principle, what is the volume of the hemisphere?

   • A) $144\pi\text{ cm}^3$
   • B) $216\pi\text{ cm}^3$
   • C) $288\pi\text{ cm}^3$
   • D) $72\pi\text{ cm}^3$
4. 4.

   Two stacks of identical cards have the same vertical height $h$, but one is arranged as a right prism with volume $V$ and the other is sheared into an oblique prism with slant height $s > h$. According to Cavalieri's principle, what is the volume of the oblique prism?

   • A) $\frac{h}{s}V$
   • B) $V$
   • C) $\frac{s}{h}V$
   • D) $\frac{s^2}{h^2}V$
5. 5.

   A right cylinder and an oblique cylinder both have a height of 15 cm, and the right cylinder has a base radius of 4 cm. If every cross section parallel to the bases at the same height has equal area in both solids, what is the volume of the oblique cylinder?

   • A) $120\pi\text{ cm}^3$
   • B) $240\pi\text{ cm}^3$
   • C) $360\pi\text{ cm}^3$
   • D) $480\pi\text{ cm}^3$

Answers and step-by-step solutions appear after you finish the test.