Question

A solid cube of side 12cm is cut into 8 cubes of equal volume. What will be the side length of the new cube? Also, find the ratios between their surface areas.

Answer 1

Hint: We solve this problem by equating the volume of the original cube with the combined volume of 8 new cubes. The formula for the volume of a cube having the side length as \['a'\] is given as
\[V={{a}^{3}}\]
After that we find the ratio of surface areas of the original cube to the new cube by using the surface area formula of a cube having side length as \['a'\] is given as
\[S.A=6{{a}^{2}}\]

Complete step by step solution:
We are given that the side length of the original cube as 12cm

Let us assume that the side length of original cube as
\[\Rightarrow A=12cm\]
Let us assume that the volume of original cube as \[{{V}_{1}}\]
We know that the formula for volume of a cube having the side length as \['a'\] is given as
\[V={{a}^{3}}\]
By using the above formula we get the volume of original cube as
\[\begin{align}
  & \Rightarrow {{V}_{1}}={{\left( 12cm \right)}^{3}} \\
 & \Rightarrow {{V}_{1}}=1728c{{m}^{3}} \\
\end{align}\]
Let us assume that the side length of new cube as \['a'\]
Let us assume that the volume of new cube as \[{{V}_{2}}\]
By using the volume formula we get the volume of new cube as
\[\Rightarrow {{V}_{2}}={{a}^{3}}\]
We are given that the original cube is divided into 8 cubes of equal volume.
By converting the above statement into mathematical equation we get
\[\Rightarrow {{V}_{1}}=8\times {{V}_{2}}\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow 1728c{{m}^{3}}=8{{a}^{3}} \\
 & \Rightarrow {{a}^{3}}=216c{{m}^{3}} \\
 & \Rightarrow a=6cm \\
\end{align}\]
Therefore the side length of the new cube formed is 6cm.
Now, let us assume that the surface area of original cube as \[{{A}_{1}}\]
We know that the formula of the surface area formula of cube having side length as \['a'\] is given as
\[S.A=6{{a}^{2}}\]
By using the above formula we get the surface area of original cube as
\[\Rightarrow {{A}_{1}}=6{{A}^{2}}.......equation(i)\]
Let us assume that the surface area of new cube as \[{{A}_{2}}\]
Now, by using the surface area of cube formula we get the surface area of new cube as
\[\Rightarrow {{A}_{2}}=6{{a}^{2}}......equation(ii)\]
Now, by dividing the equation (i) with equation (ii) we get
\[\Rightarrow \dfrac{{{A}_{1}}}{{{A}_{2}}}=\dfrac{{{A}^{2}}}{{{a}^{2}}}\]
Now, by substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow \dfrac{{{A}_{1}}}{{{A}_{2}}}=\dfrac{{{12}^{2}}}{{{6}^{2}}} \\
 & \Rightarrow \dfrac{{{A}_{1}}}{{{A}_{2}}}=\dfrac{144}{36}=\dfrac{4}{1} \\
\end{align}\]
We know that from the definition of ratios that is
\[\Rightarrow a:b=\dfrac{a}{b}\]
By using the above formula we get the ratio of surface areas as
\[\Rightarrow {{A}_{1}}:{{A}_{2}}=4:1\]
Therefore the ratio of surface areas of original cube to new cube is \[4:1\]

Note: Students may make mistakes in the calculation of side length of new cube. We are given that the original cube is divided into 8 cubes of equal volume. By converting the above statement into a mathematical equation we get
\[\Rightarrow {{V}_{1}}=8\times {{V}_{2}}\]
But students do mistake and take the mathematical equation as
\[\Rightarrow A=8\times a\]
This gives the wrong answer because they said that the original cube is divided into 8 cubes of equal volume, not equal side length.
Also, while finding the ratio of surface area we find \[{{A}_{1}}:{{A}_{2}}\] but not \[{{A}_{1}}:8\times {{A}_{2}}\]
This point needs to be taken care of.