Chapter 12.5, Problem 26WE

To determine

To show: The ratio of the lateral area of prism is $e_1^2:e_2^2$ .

Explanation of Solution

Given information:

Two spheres have slant heights $h_1$ and $h_2$ , base areas $B_1$ and $B_2$ .

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 $B_2$ .

Lateral area of right prism in term of base edge and height is $eh$ .

Lateral area of right prismis proportional to the base edge and height. Let the lateral area of cone is $L\cdot A_1$ and $L\cdot A_2$ .

Both cones are similar so the ratio of base edge and height would be similar.

  $\frac{e_1}{e_2}=\frac{h_1}{h_2}$

Ratio of lateral area of right prism is.

  $\begin{array}{c}\frac{L\cdot A_1}{L\cdot A_2}=\left(\frac{e_1{h}_1}{e_2{h}_2}\right)\\ =\left(\frac{e_1}{e_2}\times \frac{h_1}{h_2}\right)\\ =\left(\frac{e_1^2}{e_2^2}\right)\end{array}$

From the above, it is clear that the ratio of the lateral area of prismis $e_1^2:e_2^2$ .

Hence, proved.