Chapter 12.5, Problem 27WE

To determine

To show: The ratio of the volume of prism is $e_1^3:e_2^3$ .

Explanation of Solution

Given information:

Two spheres have radii $r_1$ and $r_2$ , slant heights $l_1$ and $l_2$ .

Proof:

Consider that the base of two similar right prisms are regular pentagons with base edges $e_1$ and $e_2$ , base areas $B_1$ and $B_2$ .

Let the volume of first right prism is $V_1$ and second right prism is $V_2$ .

Volume of prism is the product of base area to the height.

  $\begin{array}{c}\frac{V_1}{V_2}=\left(\frac{B_1{h}_1}{B_2{h}_2}\right)\\ =\left(\frac{e_1^2}{e_2^2}\times \frac{e_1}{e_2}\right)\\ =\left(\frac{e_1^3}{e_2^3}\right)\end{array}$

From the above, it is clear that the ratio of the volume of prismis $e_1^3:e_2^3$ .

Hence, proved.