Chapter 12, Problem 14CT

To determine

To find: The ratio of the volumes and lateral areas of a cylinder and a cone.

Answer to Problem 14CT

The volumes and lateral areas of a cylinder and a cone are $\frac{1}{3}$ and $\frac{5}{8}$ respectively.

Explanation of Solution

Given information: Radius and height of a cone and a cylinder are $3$ and $4$ respectively.

Calculation:

The formula for volume of cone is:

  $V_1=\frac{\pi }{3}{r}^{2}h$

The formula for volume of cylinder is:

  $V_2=\pi r^{2}h$

The ratio of the volumes of a cone and a cylinder is,

  $\begin{array}{c}V=\frac{V_1}{V_2}\\ =\frac{\frac{\pi }{3}{r}^{2}h}{\pi r^{2}h}\\ =\frac{1}{3}\end{array}$

The formula for the lateral area of a cone is:

  $L.A{.}_1=\pi r\sqrt{h^2+r^2}$

The formula for the lateral area of a cylinder is:

  $L.A{.}_2=2\pi rh$

The ratio of the lateral areas of a cone and a cylinder is,

  $\begin{array}{c}L.A.=\frac{L.A{.}_1}{L.A{.}_2}\\ =\frac{\pi r\sqrt{h^2+r^2}}{2\pi rh}\\ =\frac{\sqrt{h^2+r^2}}{2h}\end{array}$

Substitute $3$ for $r$ and $4$ for $h$ in the formula.

  $\begin{array}{c}L.A.=\frac{\sqrt{4^2+3^2}}{2\cdot 4}\\ =\frac{\sqrt{16+9}}{8}\\ =\frac{\sqrt{25}}{8}\\ =\frac{5}{8}\end{array}$

Therefore, the volumes and lateral areas of a cylinder and a cone are $\frac{1}{3}$ and $\frac{5}{8}$ respectively.