One side of a cube is measured as `a=4.03pm 1%`. What is its value?

Options

A. `65.45m^(3)pm1%`
B. `65.45m^(3)pm2%`
C. `65.45m^(3) pm 3%`
D. `65.45m^(3)pm4%`

Explanation

To solve the problem, we need to find the value of the volume of a cube when one side is measured as \( a = 4.03 \, \text{pm} \pm 1\% \).

### Step-by-Step Solution:

1. **Identify the Measurement and Uncertainty**:
The side length of the cube is given as \( a = 4.03 \, \text{pm} \) with an uncertainty of \( 1\% \).

2. **Calculate the Uncertainty in the Measurement**:
To find the absolute uncertainty \( \Delta a \), we calculate \( 1\% \) of \( 4.03 \, \text{pm} \):
\[ \Delta a = \frac{1}{100} \times 4.03 \, \text{pm} = 0.0403 \, \text{pm} \]

3. **Calculate the Volume of the Cube**:
The volume \( V \) of a cube is calculated using the formula:
\[ V = a^3 \]
Substituting the value of \( a \):
\[ V = (4.03 \, \text{pm})^3 = 4.03 \times 4.03 \times 4.03 \]
Performing the calculation:
\[ V \approx 65.4 \, \text{pm}^3 \]

4. **Determine the Significant Figures**:
The measurement \( 4.03 \) has three significant figures. Therefore, the volume \( V \) should also be expressed with three significant figures:
\[ V = 65.4 \, \text{pm}^3 \]

5. **Calculate the Uncertainty in Volume**:
The relative uncertainty in volume can be calculated using the formula:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta a}{a} \]
Substituting the values:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{0.0403 \, \text{pm}}{4.03 \, \text{pm}} = 3 \cdot 0.01 = 0.03 \]
Thus, the percentage uncertainty in volume is:
\[ \Delta V = 0.03 \times 100\% = 3\% \]

6. **Final Result**:
Therefore, the final value of the volume with its uncertainty is:
\[ V = 65.4 \, \text{pm}^3 \pm 3\% \]

### Final Answer:
The value of the volume is \( 65.4 \, \text{pm}^3 \pm 3\% \).