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: {Volume expansion formula}
\[\Delta V = V_0 \gamma \Delta T\]Step 2: {Volume change in terms of side length}
\[\Delta V = a^3 (3\alpha) \Delta T\]Step 3: {Determination of cube side length}
\[6a^2 = 24 \quad \Rightarrow \quad a^2 = 4 \quad \Rightarrow \quad a = 2\]Step 4: {Substitution of values}
\[\Delta V = 2^3 (3 \times 5 \times 10^{-4}) \times 10 = 1200 \times 10^{-4} \, m^3\]\[= 1200 \times 10^2 \, cm^3 = 1.2 \times 10^5 \, cm^3\]Consequently, the final answer is \( 1.2 \times 10^5 \, cm^3 \).