When the temperature of a coin is raised by `75^@C` , the coin's diameter increase by `2.3xx10^(-5)m` . If the original diameter of the coin is `1.8 xx10^(-2)` m, find the coefficient of linear expansion .

Options

A. `5.4 xx10^(-5)//""^@C`
B. `1.7 xx10^(-5)//""^@C`
C. `9.8 xx10^(-6)//""^@C`
D. `3.1 xx10^(-6)//""^@C`

Answer

To find the coefficient of linear expansion (α) of the coin given the change in diameter due to a temperature change, we can follow these steps:

### Step-by-Step Solution

1. **Identify the given values:**
- Change in temperature (ΔT) = 75°C
- Change in diameter (Δd) = 2.3 × 10^(-5) m
- Original diameter (d₀) = 1.8 × 10^(-2) m

2. **Recall the formula for linear expansion:**
The formula for linear expansion is given by:
\[ ΔL = α \cdot L₀ \cdot ΔT \]
where:
- ΔL is the change in length (or diameter in this case),
- α is the coefficient of linear expansion,
- L₀ is the original length (or diameter),
- ΔT is the change in temperature.

3. **Rearrange the formula to solve for α:**
To find α, we rearrange the formula:
\[ α = \frac{ΔL}{L₀ \cdot ΔT} \]

4. **Substitute the known values into the formula:**
Now, substitute the values we have into the rearranged formula:
\[ α = \frac{2.3 \times 10^{-5}}{(1.8 \times 10^{-2}) \cdot 75} \]

5. **Calculate the denominator:**
First, calculate the denominator:
\[ (1.8 \times 10^{-2}) \cdot 75 = 1.35 \times 10^{-1} \text{ m} \]

6. **Calculate α:**
Now, substitute the denominator back into the equation for α:
\[ α = \frac{2.3 \times 10^{-5}}{1.35 \times 10^{-1}} \approx 1.7037 \times 10^{-4} \text{ per °C} \]

7. **Round the answer:**
Rounding the value gives:
\[ α \approx 1.7 \times 10^{-5} \text{ per °C} \]

### Final Answer:
The coefficient of linear expansion (α) of the coin is approximately \(1.7 \times 10^{-5} \, \text{per °C}\).