The coefficient of linear expansion of iron is ${10}^{-5}/{}^{\circ }C$. The volume of an iron cube, $5cm$ on edge will increase by what amount if it is heated from $10°\mathrm{C}$ to $60°\mathrm{C}$?

The correct option is A

$0.1875{\mathrm{cm}}^3$

Step 1: Given Data

Coefficient of Linear Expansion $\left(\alpha \right)$ of iron is ${10}^{-5}/{}^{\circ }\mathrm{C}$

Edge $\left(l\right)$ of iron cube is $5cm$

Temperature is changed from $10°\mathrm{C}$ to $60°\mathrm{C}$

To Find: Change in volume of iron cube when heated

Step 2: Recall relation for volume expansion and relation between volume and linear expansion:

$\frac{\Delta V}{V}=\gamma \Delta T$, where $\gamma$ is the coefficient of volume expansion, $\Delta V$ is the change in volume, $V$ is the original volume and $\Delta T$ is the change in temperature.
$\gamma =3\alpha$, where $\left(\alpha \right)$is the coefficient of linear expansion.

Step 3: Calculate change in volume

Using the above two formulas, we can get,

$\frac{\Delta V}{V}=3\alpha \Delta T$

$\Delta V=3\alpha \times V\times \Delta T$

$\Delta V-3\alpha \times \left(l^3\right)\times \left(T_2-T_1\right)$

$\Delta V=3\times \left({10}^{-5}/{}^{\circ }\mathrm{C}\right)\times (5\mathrm{cm}{)}^3\times \left(60°\mathrm{C}-10°\mathrm{C}\right)$

$\Delta V=3\times \left({10}^{-5}\right)\times \left(125\right)\times \left(50\right)$

$\Delta V=0.1875{\mathrm{cm}}^3$

The change in volume will be $\Delta V=0.1875{\mathrm{cm}}^3$
Hence option A is correct.