A non-isotropic solid metal cube has coefficients of linear expansion as ` 5 xx 10^(–5) //^(@)C` along the x-axis and` 5 xx 10^(–6)//""^(@)C `along the y and the z-axis. If coefficient of volume expansion of the solid is `C xx 10^(–6) //""^(@)C` then the value of C is

Answer

To solve the problem, we need to find the coefficient of volume expansion \( \gamma \) of a non-isotropic solid metal cube given its coefficients of linear expansion along different axes.

### Step-by-step Solution:

1. **Understanding Coefficients of Linear Expansion**:
- The coefficient of linear expansion along the x-axis is given as:
\[ \alpha_x = 5 \times 10^{-5} \, ^\circ C^{-1} \]
- The coefficients of linear expansion along the y and z axes are:
\[ \alpha_y = \alpha_z = 5 \times 10^{-6} \, ^\circ C^{-1} \]

2. **Formula for Coefficient of Volume Expansion**:
- The coefficient of volume expansion \( \gamma \) can be expressed in terms of the coefficients of linear expansion as:
\[ \gamma = \alpha_x + \alpha_y + \alpha_z \]

3. **Substituting the Values**:
- Now, substitute the values of \( \alpha_x \), \( \alpha_y \), and \( \alpha_z \) into the formula:
\[ \gamma = 5 \times 10^{-5} + 5 \times 10^{-6} + 5 \times 10^{-6} \]

4. **Calculating the Coefficient of Volume Expansion**:
- First, convert all terms to the same power of ten:
\[ \gamma = 5 \times 10^{-5} + 5 \times 10^{-6} + 5 \times 10^{-6} = 5 \times 10^{-5} + 10 \times 10^{-6} \]
- Since \( 10 \times 10^{-6} = 1 \times 10^{-5} \), we can add:
\[ \gamma = 5 \times 10^{-5} + 1 \times 10^{-5} = 6 \times 10^{-5} \]

5. **Relating to the Given Coefficient**:
- According to the problem, the coefficient of volume expansion is given as:
\[ \gamma = C \times 10^{-6} \, ^\circ C^{-1} \]
- Setting the two expressions for \( \gamma \) equal gives:
\[ 6 \times 10^{-5} = C \times 10^{-6} \]

6. **Solving for C**:
- To find \( C \), rearrange the equation:
\[ C = \frac{6 \times 10^{-5}}{10^{-6}} = 6 \times 10^{1} = 60 \]

### Final Answer:
The value of \( C \) is \( 60 \).

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