Chapter 12.5, Problem 25WE

To determine

To show: The ratio of the lateral areas of cone is $r_1^2:r_2^2$ .

Explanation of Solution

Given information:

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

Proof:

Lateral areas of cone in terms of radius and slant height is $\pi rl$ . The lateral area of cone is proportional to the radius and slant height.

Let’s the lateral area of cones is $L\cdot A_1$ and $L\cdot A_2$ .

Both the cones are similar so ratio of slant heights and radius would be similar.

  $\frac{r_1}{r_2}=\frac{l_1}{l_2}$

Ratio of the lateral area of two cones.

  $\begin{array}{c}\frac{L\cdot A_1}{L\cdot A_2}=\frac{\pi r_1{l}_1}{\pi r_2{l}_2}\\ =\frac{\pi r_1}{\pi r_2}\times \frac{l_1}{l_2}\\ =\frac{r_1}{r_2}\times \frac{r_1}{r_2}\\ =\frac{r_1^2}{r_2^2}\end{array}$

From the above, it is clear that the ratio of the lateral area of conesis $r_1^2:r_2^2$ .

Hence, proved.