Question

If the length of each edge of a cube is tripled, then what is the change in its volume?

Answer 1

Hint: The cube is a kind of cuboid whose dimensions, that is, the length, breadth and height are the same. The volume of a cuboid will be the product of its dimension, that is, the product of its breadth, length and height. Since this is a cube, the value of length, breadth and height will all be the same and hence we arrive at the formula, \[V = {a^3}\] .

Formula used:
The volume of a cube is given by the formula, \[V = {a^3}\], where $a$ is the side a cube.

Complete step by step answer:
Let us start by assuming that the value of each edge of the given cube has a length of \[a\] meters. The volume of a cube can be calculated using the formula \[V = {a^3}\].If the length of the cube is tripled, then the value of the new edge of the cube would be \[a' = 3a\] meters which makes the value of the new volume of the cube to be,
\[V' = {(a')^3}\]
\[\Rightarrow V' = {(3a)^3} \]
\[\Rightarrow V'= 27{a^3}\]
The change in the volume of the cube when each edge of it is tripled will be
\[\dfrac{{V'}}{V} = \dfrac{{27{a^3}}}{{{a^3}}} \]
\[\therefore \dfrac{{V'}}{V}= 27\,times\]

Hence, if the length of each edge of a cube is tripled, the volume would be \[27\,times\] that of the original.

Note: We know that the volume of a cube is given by, \[V = {a^3}\]. It is possible that some people tend to take \[3 \times {a^3}\] instead of \[{(3a)^3}\] and get the answer with a different constant that will eventually lead to a different answer. The formulation and mathematical procedure should be carried out with caution because we often tend to make mistakes while calculating ratios and products.