Question

A solid metallic cube having total surface area $24 \, m^2$ is uniformly heated. If its temperature is increased by $10^\circ C$, calculate the increase in volume of the cube.
% Given: $\alpha = 5.0 \times 10^{-4} \, C^{-1}$

• A. $2.4 \times 10^6 \, cm^3$
• B. $1.2 \times 10^5 \, cm^3$
• C. $6.0 \times 10^4 \, cm^3$
• D. $4.8 \times 10^5 \, cm^3$

Hint:

For thermal expansion in solids, volume change is calculated using $\Delta V = V_0 \gamma \Delta T$, where $\gamma = 3\alpha$.

Answer 1

The Correct Option is B

Step 1: {Formula for volume expansion}
$$\Delta V = V_0 \gamma \Delta T$$ Step 2: {Volume relation with side length}
$$\Delta V = a^3 (3\alpha) \Delta T$$ Step 3: {Finding cube side length}
$$6a^2 = 24 \quad \Rightarrow \quad a^2 = 4 \quad \Rightarrow \quad a = 2$$ Step 4: {Substituting values}
$$\Delta V = 2^3 (3 \times 5 \times 10^{-4}) \times 10 = 1200 \times 10^{-4} \, m^3$$ $$= 1200 \times 10^2 \, cm^3 = 1.2 \times 10^5 \, cm^3$$ Thus, the correct answer is $1.2 \times 10^5 \, cm^3$.