A vernier caliper with a least count of of `0.01 cm` was used to measure diameter of cylinder as `4 cm` and a scale `(0-15cm)` with the least count of `1 mm` was used to measue a length of `5cm`. The `%` error in the measurement of volume of the cylinder is

Options

A. `3.0`
B. `4.0`
C. `5.0`
D. `2.5`

Explanation

To find the percentage error in the measurement of the volume of the cylinder, 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) of the cylinder.

### Step 2: Relate diameter to radius
Since the diameter \( d \) is given as \( 4 \, \text{cm} \), we can find the radius:
\[ r = \frac{d}{2} = \frac{4 \, \text{cm}}{2} = 2 \, \text{cm} \]

### Step 3: Determine the length of the cylinder
The length \( h \) of the cylinder is given as \( 5 \, \text{cm} \).

### Step 4: Calculate the volume of the cylinder
Substituting the values of \( r \) and \( h \) into the volume formula:
\[ V = \pi (2 \, \text{cm})^2 (5 \, \text{cm}) = \pi (4 \, \text{cm}^2)(5 \, \text{cm}) = 20\pi \, \text{cm}^3 \]

### Step 5: Calculate the percentage error in volume
To find the percentage error in the volume, we need to consider the errors in the measurements of diameter and length.

1. **Error in diameter measurement**:
- The least count of the vernier caliper is \( 0.01 \, \text{cm} \).
- The maximum possible error in diameter measurement is \( 0.01 \, \text{cm} \).
- The percentage error in diameter is calculated as:
\[ \text{Percentage error in diameter} = \left( \frac{\text{Error}}{\text{Measured value}} \right) \times 100 = \left( \frac{0.01 \, \text{cm}}{4 \, \text{cm}} \right) \times 100 = 0.25\% \]
- Since the volume depends on the square of the radius, we multiply this error by 2:
\[ \text{Total percentage error from diameter} = 2 \times 0.25\% = 0.5\% \]

2. **Error in length measurement**:
- The least count of the scale is \( 1 \, \text{mm} = 0.1 \, \text{cm} \).
- The maximum possible error in length measurement is \( 0.1 \, \text{cm} \).
- The percentage error in length is calculated as:
\[ \text{Percentage error in length} = \left( \frac{0.1 \, \text{cm}}{5 \, \text{cm}} \right) \times 100 = 2\% \]

### Step 6: Combine the percentage errors
Now, we combine the percentage errors:
\[ \text{Total percentage error in volume} = \text{Error from diameter} + \text{Error from length} = 0.5\% + 2\% = 2.5\% \]

### Final Answer
The percentage error in the measurement of the volume of the cylinder is \( 2.5\% \).

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