Question

A solid cube is cut into two cuboids exactly at the middle as shown in figure. Find the ratio of total surface area of the given cube and that of the cuboid.

A. $3:2$
B. $2:5$
C. $5:7$
D. $6:1$

Answer 1

Hint: As given in the diagram the measure of the side of a cube is $l$. So, we calculate the total surface area of the cube by using the formula $6{{a}^{2}}$ where, $a$ is the measure of the side of a cube. Then, as given in the question cube is cut into two cuboids exactly at middle so we calculate the total surface area of cuboid by using the formula $2\left( lb+bh+lh \right)\text{ c}{{\text{m}}^{2}}$. Then, we obtain a required ratio.

Complete step-by-step answer:
We have given that A solid cube is cut into two cuboids exactly at the middle as shown in figure.
We have to find the ratio of total surface area of the given cube and that of the cuboid.
As given in the figure let the measure of the side of the cube be $l$.
Then, the total surface area of the cube will be $6{{l}^{2}}$.
Now, the given cube is cut into two cuboids exactly at the middle. So, the width will be $\dfrac{l}{2}$ but the height and length will remain same i.e. $l$.
Now, the total surface area of the cuboid will be $2\left( lb+bh+lh \right)\text{ c}{{\text{m}}^{2}}$.
\[\begin{align}
  & \Rightarrow 2\left( l\times \dfrac{l}{2}+\dfrac{l}{2}\times l+l\times l \right)\text{ c}{{\text{m}}^{2}} \\
 & \Rightarrow 2\left( \dfrac{{{l}^{2}}}{2}+\dfrac{{{l}^{2}}}{2}+{{l}^{2}} \right)\text{ c}{{\text{m}}^{2}} \\
 & \Rightarrow 2\left( \dfrac{{{l}^{2}}+{{l}^{2}}+2{{l}^{2}}}{2} \right)\text{ c}{{\text{m}}^{2}} \\
 & \Rightarrow 2\left( \dfrac{4{{l}^{2}}}{2} \right)\text{ c}{{\text{m}}^{2}} \\
 & \Rightarrow 4{{l}^{2}}\text{ c}{{\text{m}}^{2}} \\
\end{align}\]
Now, the required ratio will be
Ratio of total surface area of the given cube and that of the cuboid $=\dfrac{6{{l}^{2}}}{4{{l}^{2}}}=3:2$
Option A is the correct answer.

Note: Both cube and cuboid look the same in the structure but there is a difference in both. The Cube has six square shaped faces of the equal measure but the cuboid has rectangular faces with different length, width and height. In this question the length and height of the cuboid is the same because it is cut from the cube. So, keep in the mind while solving the question.