Question
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The lateral surface of a cylinder is equal to the curved surface of a cone. If the radius will be the same. Find the ratio of the height of the cylinder and the slant height of the cone.
A)$1:3$ B) $1:4$ C) $1:2$ D) $1:5$

Answer Verified Verified
Hint: The lateral surface area of cylinder $ = 2\pi {\text{rh}}$ where r is the radius of cylinder and h is height of the cylinder and the curved surface of cone $ = \pi {\text{Rl}}$ where R is the radius of cone and l is slant height of cone. According to the question, both are equal and the radius of both shapes are also same. Put the given values and find the ratio.

Complete step by step answer:

Given, the lateral surface of a cylinder is equal to the curved surface of a cone. The radius of the cylinder is equal to the radius of the cone. We have to find the ratio of the height of the cylinder and the slant height of the cone. We know that the lateral surface area of cylinder$ = 2\pi {\text{rh}}$where r= the radius of cylinder and h= height of cylinder. Also, the curved surface area of cone$ = \pi {\text{Rl}}$where R=radius of the cone and l- slant height of the cone. Here, given r=R and according to question,
The lateral surface area of a cylinder= the curved surface area of the cone
On putting the given values we get,
$
   \Rightarrow 2\pi {\text{rh = }}\pi {\text{rl}} \\
    \\
 $
On simplifying we get,
$ \Rightarrow \dfrac{{\text{h}}}{{\text{l}}} = \dfrac{{\pi {\text{r}}}}{{2\pi {\text{r}}}} = \dfrac{1}{2}$
Hence the correct answer is ‘C’.

Note: The lateral surface area of a cylinder is the area of all sides of the cylinder excluding the area of its base and top. The curved surface area of the cone is the area of cone excluding the area of circle present in the cone. It is also called the lateral area. Here the student may get confused and could use the formula of total surface area of cylinder and total surface area of cone which are- $2\pi {\text{r}}\left( {{\text{h + r}}} \right)$ and $(\pi {\text{rl + }}\pi {{\text{r}}^2})$ respectively where the variables have usual meaning. But we are asked but the lateral surface, not the total surface.