Question

A right circular cylinder and a cone have equal bases and equal heights. If their curved surface areas are in the ratio 8:5, show that the ratio between the radius of their bases to their height is 3:4.

Answer 1

Hint:In this question, we use the formula of curved surface area of cylinder and cone. Curved surface area of cylinder is $$C.S.A=2\pi rh$$ and Curved surface area of cone is $$C.S.A=\pi rl=\pi r\sqrt{h^2+r^2}$$ .

Complete step-by-step answer:

Given, the cylinder and the cone have equal bases and equal heights.
Let the radius and height of the cylinder and cone is r and h.
Let slant height of cone is $l$
Curved surface area of cylinder is $$C.S.A=2\pi rh$$ and Curved surface area of cone is $$C.S.A=\pi rl=\pi r\sqrt{h^2+r^2}$$ .
We know the ratio of Curved surface area of cylinder and cone is 8:5.
$$\begin{gathered} \Rightarrow {\displaystyle \frac{\text{Curved surface area of cylinder}}{\text{Curved surface area of cone}}}={\displaystyle \frac{2\pi rh}{\pi rl}}={\displaystyle \frac{8}{5}} \\ \Rightarrow {\displaystyle \frac{2\pi rh}{\pi rl}}={\displaystyle \frac{8}{5}} \\ \Rightarrow {\displaystyle \frac{h}{l}}={\displaystyle \frac{4}{5}} \end{gathered}$$
Using $l=\sqrt{h^2+r^2}$
$$\Rightarrow {\displaystyle \frac{h}{\sqrt{h^2+r^2}}}={\displaystyle \frac{4}{5}}$$
Squaring both sides,
 $$\begin{gathered} \Rightarrow {\displaystyle \frac{h^2}{h^2+r^2}}={\displaystyle \frac{16}{25}} \\ \Rightarrow 25h^2=16h^2+16r^2 \\ \Rightarrow 9h^2=16r^2 \\ \Rightarrow {\displaystyle \frac{r^2}{h^2}}={\displaystyle \frac{9}{16}} \end{gathered}$$
Taking square root,
$$\begin{gathered} \Rightarrow {\displaystyle \frac{r}{h}}=\sqrt{{\displaystyle \frac{9}{16}}} \\ \Rightarrow {\displaystyle \frac{r}{h}}={\displaystyle \frac{3}{4}} \end{gathered}$$
So, the ratio between the radius of their bases to their height is 3:4.

Note: Whenever we face such types of problems we use some important points. First we take a ratio of curved surface area of cylinder and cone then use the formula of slant height $l=\sqrt{h^2+r^2}$ . So, after some calculation we will get the required answer.