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If the coefficient of cubical expansions is $x$ times of the coefficient of superficial expansion, then the value of $x$ is:
A) $2.7$
B) $2$
C) $1.5$
D) $9.5$

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Last updated date: 19th Apr 2024
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Answer
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Hint: The increase in the area and volume with the rise in the temperature is known as superficial and cubical expansion. By finding the relation between these two expansions we can solve the given question.

Formula used:
$ \Rightarrow \dfrac{\alpha }{1} = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$

Complete step by step answer:
To answer the given question, we need to understand the expansion phenomena. Whenever there is an expansion of the body due to the heating, then the body is said to be expanding and this phenomenon is known as the expansion phenomena.
The solids can undergo this phenomenon. There are totally three types of phenomenon. They are:
1. Linear expansion is the expansion that is caused due to the increase in the length of the solid. It is denoted by $\alpha $
2. Superficial expansion is the expansion that is caused due to the increase in the area of the solid. It is denoted by $\beta $
3. Cubical expansion is the expansion that is caused due to the increase in the volume of the solid. It is denoted by $\gamma $
The relation between these expansions is $\alpha ,\beta ,\gamma $. From this, it is clear that the expansions occur with the increase in the temperature.
The common relation between these expansions is:
$ \Rightarrow \dfrac{\alpha }{1} = \dfrac{\beta }{2} = \dfrac{\gamma }{3}$
It can also be represented as,
$ \Rightarrow \alpha :\beta :\gamma = 1:2:3$
Now, let us try to solve the given problem. In the question, they have given about the cubical and the superficial expansions. So, we can consider these two expansions alone. We have a relation between these two expansions. The relation is,
$ \Rightarrow \dfrac{\beta }{2} = \dfrac{\gamma }{3}$
The above equation can be written as,
$ \Rightarrow \gamma = \beta \dfrac{3}{2}........(1)$
In the question, it is given that the coefficient of the cubical expansion is increased by the $x$ times of the coefficient of superficial expansion. That is,
$ \Rightarrow \gamma = \beta x...........(2)$
We can compare equations 1 and 2. We get the answer as,
$ \Rightarrow x = \dfrac{3}{2}$
We can use division to simplify, we get,
$ \Rightarrow x = 1.5$
The value of $x$ is $1.5.$
$\therefore x = 1.5$

Hence option \[\left( C \right)\] is the correct answer.

Note: We have some basic formulae to calculate the expansions. To calculate the area expansion, we have ${A_0}(I + \beta t)$ where $\beta $ is the coefficient of the expansion. To calculate the volume expansion, we have, $\Delta V = {V_\gamma }\Delta t$ where $\gamma $ is the coefficient of volume expansion.