A circular hole in an aluminium plates is 3.115 cm in diameter at `0.000^@C` . What is its diameter when the temperature of the plate is raised to `180.0^@C` ?

Answer

To solve the problem of finding the new diameter of a circular hole in an aluminum plate when the temperature is raised, we can follow these steps:

### Step 1: Identify the initial parameters
- Initial diameter of the hole, \( d_{\text{initial}} = 3.115 \, \text{cm} \)
- Initial temperature, \( T_{\text{initial}} = 0.0 \, ^\circ C \)
- Final temperature, \( T_{\text{final}} = 180.0 \, ^\circ C \)

### Step 2: Calculate the change in temperature
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 180.0 \, ^\circ C - 0.0 \, ^\circ C = 180.0 \, ^\circ C \]

### Step 3: Use the linear expansion formula
The formula for linear expansion is given by:
\[ d_{\text{final}} = d_{\text{initial}} \times (1 + \alpha \Delta T) \]
where:
- \( \alpha \) is the coefficient of linear expansion for aluminum, which is \( 2.5 \times 10^{-5} \, \text{per Kelvin} \).

### Step 4: Substitute values into the formula
Substituting the known values into the formula:
\[ d_{\text{final}} = 3.115 \, \text{cm} \times \left(1 + (2.5 \times 10^{-5}) \times 180.0\right) \]

### Step 5: Calculate the expansion factor
First, calculate the expansion factor:
\[ \alpha \Delta T = 2.5 \times 10^{-5} \times 180.0 = 0.0045 \]
Now, add this to 1:
\[ 1 + 0.0045 = 1.0045 \]

### Step 6: Calculate the final diameter
Now, substitute back to find \( d_{\text{final}} \):
\[ d_{\text{final}} = 3.115 \, \text{cm} \times 1.0045 \approx 3.129 \, \text{cm} \]

### Conclusion
The diameter of the hole when the temperature of the plate is raised to \( 180.0 \, ^\circ C \) is approximately \( 3.129 \, \text{cm} \).

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