Question

12 solids, spheres of the same size are made by melting a solid metallic cone of base radius $1$cm and height of $48$cm. Find the radius of each sphere.

Answer 1

Hint: We will assume the radius of the solid sphere as $r$cm. We know that the volume of the solid sphere of radius $r$cm is given by $V=\dfrac{4}{3}\pi {{r}^{3}}$. From this, we will calculate the volume of the $12$ solid sphere having the same radius $r$cm by multiplying the volume of one sphere with $12$. Now in the problem, they have mentioned that these twelve solid spheres are made by melting solid metallic cones. In this case, the total volume of twelve solid spheres is equal to the volume of the cone. So we will calculate the volume of the cone by knowing formula $V=\dfrac{1}{3}\pi {{R}^{2}}h$, where $R$ is the radius of the cone and $h$ is the height of the cone. Now we will equate both the volumes to get the radius of the solid sphere.

Complete step-by-step solution
Given that,
$12$ solids, spheres of the same size are made by melting a solid metallic cone of base radius $1$cm and height of $48$cm as shown in below figure

Let the radius of one solid sphere is $r$cm as shown in the below figure

Now the volume of the solid sphere having radius $r$cm is given by ${{V}_{s}}=\dfrac{4}{3}\pi {{r}^{3}}$.
Twelve solids are made by melting a cone, so the volume of total twelve cones is $\begin{align}
  & 12\times {{V}_{s}}=12\times \dfrac{4}{3}\pi {{r}^{3}} \\
 & \Rightarrow 12\times {{V}_{s}}=16\pi {{r}^{3}} \\
\end{align}$
When we manufactured twelve solids from cones, then the volume of twelve spheres should be equal to the volume of the cone. We have given the dimensions of the cone as radius $R=1$cm and height $h=48$cm. Now the volume of the cone is given by
$\begin{align}
  & {{V}_{c}}=\dfrac{1}{3}\pi {{R}^{2}}h \\
 & \Rightarrow {{V}_{c}}=\dfrac{1}{3}\times \pi {{\left( 1 \right)}^{2}}\times 48 \\
 & \Rightarrow {{V}_{c}}=16\pi \\
\end{align}$
But
$\begin{align}
  & {{V}_{c}}=12{{V}_{s}} \\
 & \Rightarrow 16\pi =16\pi {{r}^{3}} \\
 & \Rightarrow {{r}^{3}}=1 \\
 & \Rightarrow r=1 \\
\end{align}$
$\therefore $ The radius of the solid sphere is $1$cm.

Note: There are different models for this kind of problem. Here they have mentioned that the solid spheres are formed by melting a cone, sometimes they will mention the solid spheres are manufactured by melting a cylinder of given dimensions, then we need to calculate the volume of a cylinder by using the formula $V=\pi {{R}^{2}}h$.