Chapter 12.5, Problem 24WE

To determine

To show: The ratio of the volumesis $h_1{}^3:h_2{}^3$ .

Explanation of Solution

Given information:

Two similar cones have radii $r_1$ and $r_2$ , and heights $h_1$ and $h_2$ .

Volume of cone in terms of radius and height is $\frac{1}{3}\pi r^{2}h$ .Let’s volume of the first cone is $V_1$ . The volume of the second cone is $V_2$ .

Since both the cones are similar. So, the ratio of their radius and height will be equal.

  $\frac{r_1}{r_2}=\frac{h_1}{h_2}$

Volume of cones is proportional to the cube of height, ratio the volume of two cones.

  $\begin{array}{c}\frac{V_1}{V_2}=\frac{\frac{1}{3}\pi r_1^2{h}_1}{\frac{1}{3}\pi r_2^2{h}_2}\\ =\frac{\frac{1}{3}\pi h_1}{\frac{1}{3}\pi h_2}\times \frac{r_1^2}{r_2^2}\\ =\frac{h_1}{h_2}\times \frac{h_1^2}{h_2^2}\\ =\frac{h_1^3}{h_2^3}\end{array}$

From the above, it is clear that the ratio of the volume of coneis $h_1{}^3:h_2{}^3$ .

Hence, proved.