Question

There are two cones. The curved surface area of one is twice that of the other. The slant height of the latter is twice that of the former. The ratio of their radii is
${\text{(A) 4:1}}$
${\text{(B) 4:3}}$
${\text{(C) 3:4}}$
${\text{(D) 1:4}}$

Answer 1

Hint: Here we have to find out the ratio of the radius. First, we will equate the area of both the cones and use the relations given to us to find the ratio between the lengths of the radii.

Formula used: Curved surface area of the cone, $A = \pi rl$, here, $r$ is the radius of the cone, $A$ is the curved surface area of the cone and $l$ is the slant height of the cone.

Complete step-by-step solution:
Let us assume the curved surface of the two cones be ${A_1}$ and ${A_2}$ respectively,
The slant height of the two cones are ${l_1}$ and ${l_2}$ respectively and the radii of the two cones are ${r_1}$ and ${r_2}$ respectively.
Now it is stated that the question as the curved surface area of one cone is twice the other, mathematically it can be written as:
$ \Rightarrow {A_1} = 2{A_2}$
On using the formula for the curved surface area, we get:
$ \Rightarrow \pi {r_1}{l_1} = 2\pi {r_2}{l_2}$
Since $\pi $ is common in both the places, it can be cancelled and written as:
$ \Rightarrow {r_1}{l_1} = 2{r_2}{l_2}$
Now it is given that the slant height of one cone is twice the other which means ${l_2} = 2{l_1}$
On substituting it in the equation we get:
$ \Rightarrow {r_1}{l_1} = 2{r_2}(2{l_1})$
On simplifying we get:
$ \Rightarrow {r_1}{l_1} = 4{r_2}{l_1}$
Since ${l_1}$ is common on both the sides, we can cancel them out and write it as:
$ \Rightarrow {r_1} = 4{r_2}$
On taking ${r_2}$on the left-hand side we get:
$ \Rightarrow \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{4}{1}$
Therefore, using ratios, we get:
$ \Rightarrow {r_1}:{r_2}::4:1$
Therefore, the ratio between the two radii of the cones is $4:1$

The correct option is A.

Note: The central axis of the cone is called the height of the cone and it can be used along with the radius to find the slant height using the Pythagoras theorem.
The formula for the volume of the cone is: $V = \dfrac{1}{3}\pi {r^2}h$, here $V$ is the volume, $r$ is the radius and $h$ is the height of the cone.