Question

Each side of a box made of a metal sheet in cubic shape is $\text{'}\mathrm{a}\text{'}$ at room temperature$\text{'}T\text{'}$, the coefficient of linear expansion of the metal sheet is ‘$\alpha$’. The metal sheet is heated uniformly, by a small temperature$∆T$, so that its new temperature is$T+∆T$. Calculate the increase in the volume of the metal box:

• A. $\frac{4}{3}{\mathrm{\pi a}}^3\mathrm{\alpha }∆\mathrm{T}$
• B. $4{\mathrm{\pi a}}^3\mathrm{\alpha }∆\mathrm{T}$
• C. $3a^3\alpha ∆T$
• D. $4a^3\alpha ∆T$

Answer 1

The correct option is C

$3a^3\alpha ∆T$

Step 1. Given data

The side of a box is $a$.

The coefficient of linear expansion is $\alpha$.

An increase in temperature is $∆T$

New temperature is $T+∆T$

Step 2. Finding the increase in the volume, $∆V$

The relationship between the coefficient of the linear expansion, $\alpha$ and the coefficient of volume expansion, $\gamma$ is given by

$\gamma =3\alpha$

Now, using the formula of change in the volume,

$∆V=V\times \gamma \times ∆T$

$∆V=3a^3\alpha ∆T$ [$\gamma =3\alpha$]

Hence, the correct option is $\left(C\right)$.