Chapter 12, Problem 6E

To determine

To justify:The given statement that the volume of sphere is equal to the difference between the volume of cylinder and volume of double cone with the help of cavalier’s principle.

Answer to Problem 6E

The given statement is justified that the volume of sphere is equal to the difference between the volume of cylinder and the volume of double cone.

Explanation of Solution

Formula used:

  $\begin{array}{c}\text{volume of sphere =}\frac{4}{3}\pi r^3\\ \text{volume of cylinder = }\pi r^{2}h\\ \text{volume of cone = }\frac{1}{3}\pi r^{2}h\end{array}$

Consider a diagram of sphere, the cylinder, and the double cone all have radius r and height is 2r.

  

Letxbe the number of disc above the center of each solid figure.The volume of first disc is equal to the difference of the volume of two other discs.

With this relationship, there are many discs under above solid figure. So,their total volume will be practically the same as the volume of the solid.

Now,

  $\begin{array}{c}\text{volume of sphere }=\text{ volume of cylinder}-\text{volume of double cone}\\ =\pi r^2\times 2r-2\left(\frac{1}{3}\pi r^2\times r\right)\\ =\frac{4}{3}\pi r^3\end{array}$

Therefore, the statement is justified that the volume of sphere is equal to the difference between volume of cylinder and double cone of solid.