If the length of a cylinder on heating increases by `2%`, the area of its base will increase by

Options

A. `0.5%`
B. `2%`
C. `1%`
D. `4%`

Explanation

To solve the problem of how much the area of the base of a cylinder increases when its length increases by 2%, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Linear Expansion**:
- When a material is heated, it expands. The change in length (ΔL) of the cylinder can be expressed in terms of its original length (L) and the coefficient of linear expansion (αL) as:
\[ \Delta L = L \cdot \alpha_L \cdot \Delta T \]
- Here, ΔT is the change in temperature.

2. **Given Data**:
- We know that the length of the cylinder increases by 2%. Therefore:
\[ \frac{\Delta L}{L} = 0.02 \quad \text{(which is 2%)} \]

3. **Understanding Area Expansion**:
- The area (A) of the base of the cylinder will also expand when heated. The change in area (ΔA) can be expressed as:
\[ \Delta A = A \cdot \alpha_A \cdot \Delta T \]
- The coefficient of area expansion (αA) is related to the coefficient of linear expansion (αL) by the relation:
\[ \alpha_A = 2 \alpha_L \]

4. **Relating Area Change to Length Change**:
- The fractional change in area (ΔA/A) can be related to the fractional change in length (ΔL/L):
\[ \frac{\Delta A}{A} = 2 \cdot \frac{\Delta L}{L} \]

5. **Substituting the Known Values**:
- Now, substituting the value of ΔL/L:
\[ \frac{\Delta A}{A} = 2 \cdot 0.02 = 0.04 \]

6. **Calculating the Percentage Increase**:
- To express this as a percentage, we multiply by 100:
\[ \text{Percentage Increase in Area} = 0.04 \times 100 = 4\% \]

### Final Answer:
The area of the base of the cylinder will increase by **4%**.

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