Chapter 12, Problem 19CR

To determine

To find: The ratio of volume of two similar cylinders.

Answer to Problem 19CR

Theratio of volume of two similar cylinders is 64:27.

Explanation of Solution

Given information:The two similar cylinder have lateral areas $48\pi$ and $27\pi$ .

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor. The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then

(i) The ratio of base areas, lateral areas and total areas is $a^2:b^2$ .

(ii) The ratio of base circumferences is $a:b$ .

(iii) The ratio of their volumes is $a^3:b^3$ .

The ratio of lateralareas of cylinders be $l_1$ and $l_2$ . Scale factor be $s_1$ and $s_2$ .

  $\begin{array}{c}{\left(\frac{s_1}{s_2}\right)}^2=\left(\frac{l_1}{l_2}\right)\\ =\left(\frac{48\pi }{27\pi }\right)\\ =\frac{16}{9}\end{array}$

To find the ratio of scale factor, take the square root on both sides of above equation.

  $\begin{array}{c}\sqrt{{\left(\frac{s_1}{s_2}\right)}^2}=\sqrt{\frac{16}{9}}\\ \frac{s_1}{s_2}=\frac{4}{3}\end{array}$

So, the scale factor is 4:3.

The ratio of the volumes of two similar cones from the above theorem can be calculated as:

  $\frac{V_1}{V_2}={\left(\frac{s_1}{s_2}\right)}^3$

Substitute 4 for $s_1$ and 3 for $s_2$ in the above expression.

  $\begin{array}{c}\frac{V_1}{V_2}={\left(\frac{4}{3}\right)}^3\\ =\frac{64}{27}\end{array}$

Therefore, the ratio of volume of two similar cylinders is 64:27.