Question

Find the volume of a rectangular box (cuboid) with length \[2ab\], breadth $3ac$ and height $2ac$.
A) $12{a^2}b{c^2}$
B) $12{a^3}b{c^2}$
C) $12{a^2}bc$
D) $2ab + 3ac + 2ac$

Answer 1

Hint: Since all the three dimensions of cuboid are already given, use the formula of volume of cuboid i.e. $V = L \times B \times H$, where $L,{\text{ }}B,{\text{ and }}H$ are length, breadth and height respectively. Put the value and find the answer.

Complete step-by-step answer:
According to the question, the length, breadth and height of the cuboid are given as \[2ab\], $3ac$ and $2ac$ respectively.
From this we have:
$
   \Rightarrow L = 2ab \\
   \Rightarrow B = 3ac \\
   \Rightarrow H = 2ac
 $
We have to determine the volume of the box. The box is rectangular so it is cuboidal in shape.
We know that the volume of a cuboid is given by the formula:
$ \Rightarrow V = L \times B \times H$
Putting all the above values in this formula, we’ll get:
$ \Rightarrow V = 2ab \times 3ac \times 2ac$
Simplifying this further, we’ll get:
$
   \Rightarrow V = 2 \times 3 \times 2 \times {a^3}b{c^2} \\
   \Rightarrow V = 12{a^3}b{c^2}
 $
Thus the volume of the rectangular box is $12{a^3}b{c^2}$.

Option B is the correct answer.

Additional Information: In cuboid, all the six of its faces are rectangular in shape i.e. the three dimensions (length, breadth and height) are of different lengths. So if we have to calculate the total surface area of a cuboid, it will be the sum of the areas of all the six rectangular faces. This can be shown by the formula:
$ \Rightarrow S = 2\left( {LB + BH + LH} \right)$, where $L,{\text{ }}B,{\text{ and }}H$ are length, breadth and height respectively.

Note: If all the three dimensions of a cuboid are the same then it no longer remains a cuboid. Rather it is a cube and all of its six faces are square. The formula of volume and total surface area of cube is given as:
$
   \Rightarrow V = {a^3} \\
   \Rightarrow S = 6{a^2}
 $
$a$ is the edge length of the cube.