Question

A right circular cone and a sphere have equal volumes. If the cone has radius x and height 2x, calculate the radius of the sphere.
(a) x
(b) $\dfrac{x}{\sqrt[3]{2}}$
(c) $\sqrt[3]{2}$
(d) $\dfrac{1}{\sqrt[3]{2}}$
(e) $\sqrt[3]{2x}$

Answer 1

Hint: Assume the radius of the sphere as r. Apply the formula of volume of a right circular cone given as ${{V}_{cone}}=\dfrac{1}{3}\pi {{R}^{2}}H$, where R is the radius of the cone and H is the height of the cone. Substitute R and H in terms of x provided in the question. Now, find the volume of the sphere by applying the formula ${{V}_{sphere}}=\dfrac{4}{3}\pi {{r}^{3}}$. Equate the two-volume relations and find the value of r.

Complete step-by-step solution:
Here, we have been provided with the condition that the volume of a cone of a radius x and height 2x is equal to the volume of a sphere. We have been asked to calculate the radius of the sphere.
Now let us assume the radius of the sphere as r. So, we have,

The above figure represents a right circular cone in which we have assumed the radius as R and height as H. So, we have,
H = 2x , R = x
Here, we know that the volume of a right circular cone is given by the relation ${{V}_{cone}}=\dfrac{1}{3}\pi {{R}^{2}}H$, so substituting the values of R and H in terms of x in the above volume relation, we get,
$\begin{align}
  & \Rightarrow {{V}_{cone}}=\dfrac{1}{3}\pi \times {{x}^{2}}\times 2x \\
 & \Rightarrow {{V}_{cone}}=\dfrac{2{{x}^{3}}\pi }{3}\ldots \ldots \ldots \left( i \right) \\
\end{align}$
Now let us draw a diagram of a sphere. So we have

In the above figure, we have assumed a sphere of radius r. Now, applying the formula for volume of a sphere, we get
$\Rightarrow {{V}_{sphere}}=\dfrac{4}{3}\pi {{r}^{3}}\ldots \ldots \ldots \left( ii \right)$
Now, it is given that the volume of the cone and the sphere are equal, so equating the two volume relations, we get,
$\begin{align}
  & \Rightarrow {{V}_{cone}}={{V}_{sphere}} \\
 & \Rightarrow \dfrac{2{{x}^{3}}\pi }{3}=\dfrac{4}{3}\pi {{r}^{3}} \\
\end{align}$
Cancelling the common factors, we get
\[\begin{align}
  & \Rightarrow {{x}^{3}}=2{{r}^{3}} \\
 & \Rightarrow {{r}^{3}}=\dfrac{{{x}^{3}}}{2} \\
\end{align}\]
Taking cube root on both sides, we get
$\Rightarrow r=\dfrac{x}{\sqrt[3]{2}}$
Hence, option (b) is the correct answer.

Note: One may note that here you must remember the formula of the volume of the right circular cone and sphere to solve the question. Apart from these, the volume relations of basic 3-D shapes like cube, cuboid, cylinder, hemisphere, etc. must be remembered because any two shapes can be provided. Now, in the above question it is given that the volume of the cone and the sphere are equal so do not think that their surface areas will also be the same otherwise you will get the wrong answer by equating the expressions of surface areas.