A metal rod of 30 cm length expands by 0.075 cm when its temperature is raised from `0^@C` to `100^@C`. Another rod of a different metal of length 45 cm expands by 0.045 cm for the same rise in temperature. A composite rod C made by joining A and B end to end expands by 0.040 cm when its length is 45 cm and it is heated from `0^@C` to `50^@C`. Find the length of each portion of the composite rod.

Explanation

To solve the problem step by step, we will first determine the coefficients of linear expansion for each rod, and then we will use these values to find the lengths of the portions of the composite rod.

### Step 1: Calculate the coefficient of linear expansion for rod A

Given:
- Length of rod A, \( L_A = 30 \, \text{cm} \)
- Expansion of rod A, \( \Delta L_A = 0.075 \, \text{cm} \)
- Temperature change, \( \Delta T = 100 - 0 = 100 \, \text{°C} \)
...