Determine the lengths of an iron rod and copper ruler at `0^@` C if the difference in their lengths at `50^@`C and `450^@`C is the same and is equal to 2 cm. the coefficient of linear expansion of iron`=12xx10^(-6)//K` and that of copper`=17xx10^(-6)//K`.

Options

A. 20.06m, 20.08m
B. 21.0 m, 7.00 m
C. 60.30 cm, 10.10 m
D. 11.09 m, 15.10 m

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We need to find the initial lengths of an iron rod and a copper ruler at 0°C, given that the difference in their lengths at 50°C and 450°C is the same and equal to 2 cm. We also have the coefficients of linear expansion for iron and copper.

### Step 2: Define the variables
Let:
- \( L_i \) = initial length of the iron rod at 0°C
- \( L_c \) = initial length of the copper ruler at 0°C
- \( \alpha_i = 12 \times 10^{-6} \, \text{K}^{-1} \) (coefficient of linear expansion for iron)
- \( \alpha_c = 17 \times 10^{-6} \, \text{K}^{-1} \) (coefficient of linear expansion for copper)

### Step 3: Write the length equations
The length of the iron rod at temperature \( T \) is given by:
\[ L_i(T) = L_i(0) \left(1 + \alpha_i T\right) \]
The length of the copper ruler at temperature \( T \) is given by:
\[ L_c(T) = L_c(0) \left(1 + \alpha_c T\right) \]

### Step 4: Calculate lengths at 50°C
At \( T_1 = 50°C \):
\[ L_i(50) = L_i \left(1 + \alpha_i \cdot 50\right) \]
\[ L_c(50) = L_c \left(1 + \alpha_c \cdot 50\right) \]

### Step 5: Calculate lengths at 450°C
At \( T_2 = 450°C \):
\[ L_i(450) = L_i \left(1 + \alpha_i \cdot 450\right) \]
\[ L_c(450) = L_c \left(1 + \alpha_c \cdot 450\right) \]

### Step 6: Set up the equations for the length differences
The difference in lengths at 50°C is:
\[ \Delta L_1 = L_i(50) - L_c(50) = L_i \left(1 + \alpha_i \cdot 50\right) - L_c \left(1 + \alpha_c \cdot 50\right) \]
The difference in lengths at 450°C is:
\[ \Delta L_2 = L_i(450) - L_c(450) = L_i \left(1 + \alpha_i \cdot 450\right) - L_c \left(1 + \alpha_c \cdot 450\right) \]

### Step 7: Set the differences equal to each other
Given that the differences are equal:
\[ \Delta L_1 = \Delta L_2 = 2 \, \text{cm} \]
So we can write:
\[ L_i \left(1 + \alpha_i \cdot 50\right) - L_c \left(1 + \alpha_c \cdot 50\right) = L_i \left(1 + \alpha_i \cdot 450\right) - L_c \left(1 + \alpha_c \cdot 450\right) \]

### Step 8: Simplify the equation
Expanding both sides:
\[ L_i + L_i \alpha_i \cdot 50 - L_c - L_c \alpha_c \cdot 50 = L_i + L_i \alpha_i \cdot 450 - L_c - L_c \alpha_c \cdot 450 \]
Cancelling \( L_i \) and \( L_c \) from both sides:
\[ L_i \alpha_i \cdot 50 - L_c \alpha_c \cdot 50 = L_i \alpha_i \cdot 450 - L_c \alpha_c \cdot 450 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ L_i \alpha_i (50 - 450) = L_c \alpha_c (450 - 50) \]
\[ L_i \alpha_i (-400) = L_c \alpha_c (400) \]
This simplifies to:
\[ L_i \alpha_i = -L_c \alpha_c \]

### Step 10: Substitute the values
Substituting the coefficients:
\[ L_i \cdot 12 \times 10^{-6} = -L_c \cdot 17 \times 10^{-6} \]
This gives:
\[ L_i = \frac{17}{12} L_c \]

### Step 11: Use the length difference
Using the given difference in lengths:
\[ L_i(450) - L_c(450) = 2 \]
Substituting for \( L_i \):
\[ \frac{17}{12} L_c (1 + 450 \cdot 12 \times 10^{-6}) - L_c (1 + 450 \cdot 17 \times 10^{-6}) = 2 \]
Solving this will yield the values of \( L_i \) and \( L_c \).

### Final Calculation
After solving the above equation, we find:
- \( L_c \approx 4.8 \, \text{cm} \)
- \( L_i \approx 6.8 \, \text{cm} \)

### Conclusion
The lengths of the iron rod and copper ruler at 0°C are approximately \( 6.8 \, \text{cm} \) and \( 4.8 \, \text{cm} \) respectively.