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 1: Understand the relationship between length and area
The area of the base of a cylinder (which is circular) is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the base. The length of the cylinder does not directly affect the area of the base, but if the cylinder is heated, it may expand, affecting the radius.

### Step 2: Relate the change in length to the change in radius
When the length of the cylinder increases by 2%, we can denote this change as:
\[ \frac{\Delta L}{L} = 0.02 \]
where \( \Delta L \) is the change in length and \( L \) is the original length.

### Step 3: Use the relationship between area and radius
Since the area \( A \) is proportional to the square of the radius, we can express the change in area \( \Delta A \) in terms of the change in radius \( \Delta r \):
\[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r} \]
This means that the fractional change in area is twice the fractional change in radius.

### Step 4: Relate change in radius to change in length
If the length increases, we can assume the radius also increases due to thermal expansion. For small changes, we can approximate:
\[ \frac{\Delta r}{r} \approx \frac{\Delta L}{L} \]
Thus, substituting this into the area change formula gives:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta L}{L} \]

### Step 5: Calculate the percentage change in area
Substituting the known change in length:
\[ \frac{\Delta A}{A} = 2 \times 0.02 = 0.04 \]
To express this as a percentage, we multiply by 100:
\[ \text{Percentage change in area} = 0.04 \times 100 = 4\% \]

### Conclusion
The area of the base of the cylinder will increase by **4%** when the length increases by 2%.

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