The dimensions of a rectangular block measured with a vernier callipers having least count of `0.1mm` is `5mmxx10mmxx5mm`. The maximum percentage error in measurement of volume of the blcok is

Options

A. 0.05
B. 0.1
C. 0.15
D. 0.2

Answer

To find the maximum percentage error in the measurement of the volume of a rectangular block, we can follow these steps:

### Step 1: Understand the formula for volume
The volume \( V \) of a rectangular block is given by the formula:
\[ V = L \times B \times H \]
where \( L \) is the length, \( B \) is the breadth, and \( H \) is the height.

### Step 2: Identify the dimensions and least count
From the problem, the dimensions of the block are:
- Length \( L = 5 \, \text{mm} \)
- Breadth \( B = 10 \, \text{mm} \)
- Height \( H = 5 \, \text{mm} \)

The least count of the vernier calipers is \( 0.1 \, \text{mm} \).

### Step 3: Calculate the absolute errors
The absolute error in each measurement is equal to the least count. Therefore:
- \( \Delta L = 0.1 \, \text{mm} \)
- \( \Delta B = 0.1 \, \text{mm} \)
- \( \Delta H = 0.1 \, \text{mm} \)

### Step 4: Calculate the relative errors
The relative error in each dimension can be calculated using the formula:
\[ \text{Relative Error} = \frac{\Delta x}{x} \]
where \( \Delta x \) is the absolute error and \( x \) is the measured value.

Calculating for each dimension:
- For length:
\[ \frac{\Delta L}{L} = \frac{0.1}{5} = 0.02 \]
- For breadth:
\[ \frac{\Delta B}{B} = \frac{0.1}{10} = 0.01 \]
- For height:
\[ \frac{\Delta H}{H} = \frac{0.1}{5} = 0.02 \]

### Step 5: Calculate the total relative error in volume
The total relative error in volume is the sum of the relative errors in length, breadth, and height:
\[ \text{Total Relative Error} = \frac{\Delta L}{L} + \frac{\Delta B}{B} + \frac{\Delta H}{H} \]
Substituting the values:
\[ \text{Total Relative Error} = 0.02 + 0.01 + 0.02 = 0.05 \]

### Step 6: Convert to percentage error
To find the percentage error, multiply the total relative error by 100:
\[ \text{Percentage Error} = 0.05 \times 100 = 5\% \]

### Final Answer
The maximum percentage error in the measurement of the volume of the block is \( 5\% \).

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