A copper plate has a length of 0.1 m and a width of 0.12 m at `25^@C` . The plate is uniformly heated to `175^@C`. If the liner expansion coefficient for copper is `1.7 xx10^(-5)//""^@C` what is the change in the area of the plate as a result of the increase in temperature ?

Options

A. `2.6 xx10^(-5)m^2`
B. `6.1 xx10^(-5)m^2`
C. `3.2 xx10^(-6) m^2`
D. `4.9 xx10^(-7)m^2`

Answer

To solve the problem of finding the change in the area of a copper plate due to temperature increase, we can follow these steps:

### Step 1: Identify the given values
- Length of the copper plate (L) = 0.1 m
- Width of the copper plate (W) = 0.12 m
- Initial temperature (T1) = 25°C
- Final temperature (T2) = 175°C
- Coefficient of linear expansion for copper (α) = 1.7 x 10^(-5) /°C

### Step 2: Calculate the original area (A)
The original area of the plate can be calculated using the formula:
\[ A = L \times W \]
Substituting the values:
\[ A = 0.1 \, \text{m} \times 0.12 \, \text{m} = 0.012 \, \text{m}^2 \]

### Step 3: Calculate the change in temperature (ΔT)
The change in temperature can be calculated as:
\[ ΔT = T2 - T1 = 175°C - 25°C = 150°C \]

### Step 4: Relate area expansion coefficient (β) to linear expansion coefficient (α)
The area expansion coefficient (β) is related to the linear expansion coefficient (α) by the formula:
\[ β = 2α \]
Substituting the value of α:
\[ β = 2 \times 1.7 \times 10^{-5} = 3.4 \times 10^{-5} /°C \]

### Step 5: Calculate the change in area (ΔA)
The change in area can be calculated using the formula:
\[ ΔA = A \times β \times ΔT \]
Substituting the values:
\[ ΔA = 0.012 \, \text{m}^2 \times 3.4 \times 10^{-5} /°C \times 150°C \]

### Step 6: Perform the calculation
Calculating ΔA:
\[ ΔA = 0.012 \times 3.4 \times 10^{-5} \times 150 \]
\[ ΔA = 0.012 \times 5.1 \times 10^{-3} \]
\[ ΔA = 6.12 \times 10^{-5} \, \text{m}^2 \]

### Final Answer
The change in the area of the plate as a result of the increase in temperature is:
\[ ΔA = 6.12 \times 10^{-5} \, \text{m}^2 \]

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