Each side of a box made of metal sheet in cubic shape is 'a' at room temperature 'T', the coefficient of linear expansion of the metal sheet is '`alpha`'. The metal sheet is heated uniformly, by a small temperature `Delta T` , so that its new temperature is `T + Delta T` . Calculate the increase in the volume of the metal box.

Options

A. `3a^3 alpha Delta T`
B. `4a^2 alpha Delta T`
C. `4pi a^^3 alpha Delta T`
D. `4/3 pi a^3 alpha Delta T`

Answer

To find the increase in the volume of a cubic box made of a metal sheet when heated, we can follow these steps:

### Step 1: Understand the initial conditions
The box is a cube with each side of length \( a \) at room temperature \( T \). The volume \( V \) of the cube can be calculated using the formula:
\[ V = a^3 \]

### Step 2: Determine the change in dimensions due to heating
When the temperature of the metal sheet is increased by \( \Delta T \), each linear dimension of the cube will expand. The change in length \( \Delta a \) for each side due to thermal expansion can be calculated using the formula for linear expansion:
\[ \Delta a = a \cdot \alpha \cdot \Delta T \]
where \( \alpha \) is the coefficient of linear expansion.

### Step 3: Calculate the new length of each side
The new length of each side \( a' \) after heating will be:
\[ a' = a + \Delta a = a + a \cdot \alpha \cdot \Delta T = a(1 + \alpha \Delta T) \]

### Step 4: Calculate the new volume
The new volume \( V' \) of the cube after heating can be calculated as:
\[ V' = (a')^3 = \left(a(1 + \alpha \Delta T)\right)^3 \]
Expanding this expression gives:
\[ V' = a^3(1 + \alpha \Delta T)^3 \]

### Step 5: Use the binomial approximation
For small values of \( \alpha \Delta T \), we can use the binomial approximation:
\[ (1 + x)^n \approx 1 + nx \quad \text{for small } x \]
Thus,
\[ (1 + \alpha \Delta T)^3 \approx 1 + 3\alpha \Delta T \]
So, we have:
\[ V' \approx a^3(1 + 3\alpha \Delta T) = a^3 + 3a^3\alpha \Delta T \]

### Step 6: Calculate the increase in volume
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V' - V = (a^3 + 3a^3\alpha \Delta T) - a^3 = 3a^3\alpha \Delta T \]

### Final Answer
Thus, the increase in the volume of the metal box is:
\[ \Delta V = 3a^3\alpha \Delta T \]