Chapter 7.3, Problem 51E

To determine

To Explain: The reason for the solid has the same volume as a right circular cone.

Answer to Problem 51E

The volume of solid and cone are same.

Explanation of Solution

Given information: The right circular cone has the radius $3$ and the height $12$ . The solid lie between the planes that are perpendicular to the x-axis.

Calculation:

  

According to the Cavalieri’s principle, when a cone is sliced from various sections then, the radius of cone sliced from each section is same and is equal to some constant (p) multiply by height (h) from the apex(y).

Hence, area is;

  $A=\pi {\left(yp\right)}^2$

Now, volume can be find out as:

  $\begin{array}{l}V={\displaystyle \underset{0}{\overset{h}{\int }}A.dy}\\ V={\displaystyle \underset{0}{\overset{h}{\int }}\pi y^2{p}^{2}dy}\\ V=\frac{\pi p^2{y}^3}{3}\end{array}$

But in the cylinder, cone is enclosed having radius R at $y=h$ . Therefore, it can be said that:

  $R=hp$

Substitute in the above formula:

  $V=\frac{\pi h^3{p}^2}{3}=\frac{\pi R^{2}h}{3}$

Similarly, in case of cone with the radius $3$ and the height $12$ .The volume is:

  $V=\frac{\pi \times 12\times 3^2}{3}$

Hence, the volume of a solid is equal to the volume of cone having radius $3$ and the height $12$ .