The length of a uniform rod is 100.0 cm and radius is 1.00 cm if length is measured with a meter rod having least count 1mm and radus is measured with vernier callipers having least count 0.1 mm the percentage error in calculated volume of cylinder is

Options

A. `2.1%`
B. 0.03
C. `2.01%`
D. `3.2%`

Answer

To solve the problem of calculating the percentage error in the volume of a cylinder given the measurements of length and radius, we will follow these steps:

### Step 1: Identify the given values
- Length of the rod (L) = 100.0 cm
- Radius of the rod (R) = 1.0 cm
- Least count of the meter rod = 1 mm = 0.1 cm
- Least count of the Vernier calipers = 0.1 mm = 0.01 cm

### Step 2: Determine the absolute errors
- The absolute error in length (ΔL) = Least count of meter rod = 0.1 cm
- The absolute error in radius (ΔR) = Least count of Vernier calipers = 0.01 cm

### Step 3: Write the measurements with errors
- Length (L) = 100.0 ± 0.1 cm
- Radius (R) = 1.0 ± 0.01 cm

### Step 4: Calculate the formula for the volume of a cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi R^2 L \]

### Step 5: Calculate the relative error in volume
The relative error in volume can be calculated using the formula:
\[ \frac{\Delta V}{V} = 2 \frac{\Delta R}{R} + \frac{\Delta L}{L} \]

### Step 6: Substitute the values into the relative error formula
- Substitute ΔR = 0.01 cm, R = 1.0 cm, ΔL = 0.1 cm, and L = 100.0 cm into the formula:
\[ \frac{\Delta V}{V} = 2 \left( \frac{0.01}{1.0} \right) + \left( \frac{0.1}{100.0} \right) \]

### Step 7: Calculate each term
- First term: \( 2 \left( \frac{0.01}{1.0} \right) = 0.02 \)
- Second term: \( \frac{0.1}{100.0} = 0.001 \)

### Step 8: Add the relative errors
\[ \frac{\Delta V}{V} = 0.02 + 0.001 = 0.021 \]

### Step 9: Convert relative error to percentage error
To find the percentage error, multiply the relative error by 100:
\[ \text{Percentage Error} = 0.021 \times 100 = 2.1\% \]

### Final Answer
The percentage error in the calculated volume of the cylinder is **2.1%**.

---