The length of a wire is measured with a metre scale having least count `1mm`. Its diameter is measued with a vernier calipers of least count `0.1mm`. Given that length and diameter of the wire are measured as `5cm` and `4mm`, the percentage error in the calculated value of volume of the wire will be

Options

A. 0.03
B. 0.05
C. 0.07
D. 0.02

Answer

To find the percentage error in the calculated value of the volume of the wire, we will follow these steps:

### Step 1: Understand the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length in this case).

### Step 2: Convert diameter to radius
Given that the diameter \( d \) of the wire is measured as \( 4 \, \text{mm} \), we can find the radius \( r \) as:
\[ r = \frac{d}{2} = \frac{4 \, \text{mm}}{2} = 2 \, \text{mm} \]

### Step 3: Convert measurements to consistent units
Since the length \( h \) is given as \( 5 \, \text{cm} \), we convert it to millimeters:
\[ h = 5 \, \text{cm} = 50 \, \text{mm} \]

### Step 4: Calculate the volume
Now we can calculate the volume using the radius and height:
\[ V = \pi (2 \, \text{mm})^2 (50 \, \text{mm}) = \pi (4 \, \text{mm}^2)(50 \, \text{mm}) = 200\pi \, \text{mm}^3 \]

### Step 5: Determine the least counts and calculate the absolute errors
- The least count for the diameter measurement is \( 0.1 \, \text{mm} \).
- The least count for the length measurement is \( 1 \, \text{mm} \).

The absolute error in diameter \( \Delta d \) is:
\[ \Delta d = 0.1 \, \text{mm} \]

The absolute error in length \( \Delta h \) is:
\[ \Delta h = 1 \, \text{mm} \]

### Step 6: Calculate the percentage error in diameter and length
The percentage error in diameter is:
\[ \text{Percentage error in } d = \frac{\Delta d}{d} \times 100 = \frac{0.1 \, \text{mm}}{4 \, \text{mm}} \times 100 = 2.5\% \]

The percentage error in length is:
\[ \text{Percentage error in } h = \frac{\Delta h}{h} \times 100 = \frac{1 \, \text{mm}}{50 \, \text{mm}} \times 100 = 2\% \]

### Step 7: Calculate the total percentage error in volume
The formula for the percentage error in volume \( V \) is given by:
\[ \text{Percentage error in } V = 2 \times \text{Percentage error in } d + \text{Percentage error in } h \]
Substituting the values we calculated:
\[ \text{Percentage error in } V = 2 \times 2.5\% + 2\% = 5\% + 2\% = 7\% \]

### Final Answer
Thus, the percentage error in the calculated value of the volume of the wire is:
\[ \boxed{7\%} \]