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.
\[
   \Rightarrow \dfrac{{{\text{Curved surface area of cylinder}}}}{{{\text{Curved surface area of cone}}}} = \dfrac{{2\pi rh}}{{\pi rl}} = \dfrac{8}{5} \\
   \Rightarrow \dfrac{{2\pi rh}}{{\pi rl}} = \dfrac{8}{5} \\
   \Rightarrow \dfrac{h}{l} = \dfrac{4}{5} \\
\]
Using $l = \sqrt {{h^2} + {r^2}} $
\[ \Rightarrow \dfrac{h}{{\sqrt {{h^2} + {r^2}} }} = \dfrac{4}{5}\]
Squaring both sides,
 \[
   \Rightarrow \dfrac{{{h^2}}}{{{h^2} + {r^2}}} = \dfrac{{16}}{{25}} \\
   \Rightarrow 25{h^2} = 16{h^2} + 16{r^2} \\
   \Rightarrow 9{h^2} = 16{r^2} \\
   \Rightarrow \dfrac{{{r^2}}}{{{h^2}}} = \dfrac{9}{{16}} \\
 \]
Taking square root,
\[
   \Rightarrow \dfrac{r}{h} = \sqrt {\dfrac{9}{{16}}} \\
   \Rightarrow \dfrac{r}{h} = \dfrac{3}{4} \\
 \]
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.