Question

Two cubes of each volume \[\dfrac{250}{3}\pi \text{c}{{\text{m}}^{3}}\] are melted and recast in the form of sphere. Find the total surface area of the sphere.

Answer 1

Hint: Here we will calculate the total volume of two cubes when they are melted by multiplying the given volume of cube with $2$ since we have melted $2$ cubes only. Now we will get the total volume after melting two cubes, now the two cubes are recast in the form of a sphere. So, there is no change in the volume of the sphere and the volume of melted two cubes. We will assume the radius of sphere as $r$ and we will calculate the volume of sphere with radius $r$ by using the formula $4\pi \dfrac{{{r}^{3}}}{3}$and then we will equate it to the volume of melted two cubes to get the value of radius of sphere. After getting the radius of the sphere we can find the surface area of the sphere by using the formula $A=4\pi {{r}^{2}}$.

Complete step-by-step answer:
Given that, the volume of one cube is ${{V}_{c}}=\dfrac{250}{3}\pi $
Now the volume of Two cubes is given by
$\begin{align}
  & {{V}_{m}}=2\times {{V}_{c}} \\
 & {{V}_{m}}=2\times \dfrac{250}{3}\pi \\
 & {{V}_{m}}=\dfrac{500}{3}\pi \\
\end{align}$
Let the radius of the sphere formed by the melting of two cubes is $r\text{ cm}$.

Now the volume of the sphere with radius $r\text{ cm}$ is given by
${{V}_{s}}=\dfrac{4}{3}\pi {{r}^{3}}$
When we melted the two cubes and recasted them into a sphere, there would be no change in the volume of the sphere and the volumes of melted cubes. Therefore,
${{V}_{s}}={{V}_{m}}$
Substituting the values of ${{V}_{s}}$ and ${{V}_{m}}$ in the above equation, then we will get
$\begin{align}
  & {{V}_{s}}={{V}_{m}} \\
 & \Rightarrow \dfrac{4}{3}\pi {{r}^{3}}=\dfrac{500}{3}\pi \\
\end{align}$
Multiplying the above equation with $\dfrac{3}{\pi }$, then we will get
$\begin{align}
  & \dfrac{4}{3}\pi {{r}^{3}}\left( \dfrac{3}{\pi } \right)=\dfrac{500}{3}\pi \left( \dfrac{3}{\pi } \right) \\
 & 4{{r}^{3}}=500 \\
\end{align}$
Dividing the above equation with $4$, then we will get
$\begin{align}
  & \dfrac{4{{r}^{3}}}{4}=\dfrac{500}{4} \\
 & {{r}^{3}}=125 \\
 & r=\sqrt[3]{125} \\
 & r=5cm \\
\end{align}$
Now the radius of the sphere is $5cm$.
Hence the surface of area of the sphere of radius $r=5cm$ is given by
$\begin{align}
  & A=4\pi {{r}^{2}} \\
 & A=4\pi {{\left( 5 \right)}^{2}} \\
 & A=4\pi \times 25 \\
 & A=100\pi \text{ c}{{\text{m}}^{2}} \\
\end{align}$
Hence the surface area of the Sphere is $100\pi c{{m}^{2}}$.

So, the correct answer is “Option A”.

Note: We can also find the surface area of the sphere by using the relation between surface area and volume and radius of the sphere i.e. $\dfrac{A}{{{V}_{s}}}=\dfrac{3}{r}$.
Substituting the values of ${{V}_{s}}={{V}_{m}}=\dfrac{500}{3}\pi c{{m}^{3}}$, $r=5cm$ in the above equation, then we will get
$\begin{align}
  & A=\dfrac{3}{r}\times {{V}_{s}} \\
 & A=\dfrac{3}{r}\times {{V}_{m}} \\
 & A=\dfrac{3}{5}\times \dfrac{500}{3}\pi \\
 & A=100\pi c{{m}^{2}} \\
\end{align}$
From both the methods we got the same answer.