The length of a rectangular plate is measured as 10 cm by a vernier scale of least count 0.01 cm and its breadth as 5 cm by the same scale. The percentage error in area is

Options

A. `0.1%`
B. `0.3 %`
C. `0.01%`
D. `0.05%`

Answer

To find the percentage error in the area of a rectangular plate, we can follow these steps:

### Step 1: Understand the formula for area
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{Length} \times \text{Breadth} \]

### Step 2: Identify the measurements
From the problem, we have:
- Length \( L = 10 \) cm
- Breadth \( B = 5 \) cm

### Step 3: Determine the least count
The least count of the vernier scale is given as:
\[ \text{Least Count} = 0.01 \text{ cm} \]

### Step 4: Calculate the absolute errors in measurements
The absolute error in length \( \Delta L \) and breadth \( \Delta B \) can be taken as the least count:
\[ \Delta L = 0.01 \text{ cm} \]
\[ \Delta B = 0.01 \text{ cm} \]

### Step 5: Calculate the relative errors
The relative error in length \( \frac{\Delta L}{L} \) and breadth \( \frac{\Delta B}{B} \) can be calculated as follows:
\[ \frac{\Delta L}{L} = \frac{0.01}{10} = 0.001 \]
\[ \frac{\Delta B}{B} = \frac{0.01}{5} = 0.002 \]

### Step 6: Calculate the total relative error in area
The total relative error in area \( \frac{\Delta A}{A} \) is given by the sum of the relative errors in length and breadth:
\[ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} \]
Substituting the values:
\[ \frac{\Delta A}{A} = 0.001 + 0.002 = 0.003 \]

### Step 7: Convert relative error to percentage error
To find the percentage error, we multiply the relative error by 100:
\[ \text{Percentage Error} = \frac{\Delta A}{A} \times 100 = 0.003 \times 100 = 0.3\% \]

### Final Result
The percentage error in the area of the rectangular plate is:
\[ \text{Percentage Error} = 0.3\% \]