The side of a cube is `(2.00 pm 0.01)` cm. The volume and surface area of cube are respectively :-

Options

A. `(8.00 pm 0.12) cm^(3),(24.0 pm 0.24) cm^(2)`
B. `(8.00 pm 0.01)cm^(3),(24.0 pm 0.01)cm^(2)`
C. `(8.00 pm 0.04)cm^(3),(24.0 pm 0.06)cm^(2)`
D. `(8.00 pm 0.03)cm^(3),(24.0 pm 0.02)cm^(2)`

Explanation

To solve the problem, we need to calculate the volume and surface area of a cube given the side length and its uncertainty.

### Step-by-Step Solution:

1. **Identify the given values:**
- Side length of the cube, \( A = 2.00 \, \text{cm} \)
- Uncertainty in side length, \( \Delta A = 0.01 \, \text{cm} \)

2. **Calculate the volume of the cube:**
- The formula for the volume \( V \) of a cube is given by:
\[ V = A^3 \]
- Substituting the value of \( A \):
\[ V = (2.00 \, \text{cm})^3 = 8.00 \, \text{cm}^3 \]

3. **Calculate the surface area of the cube:**
- The formula for the surface area \( S \) of a cube is given by:
\[ S = 6A^2 \]
- Substituting the value of \( A \):
\[ S = 6 \times (2.00 \, \text{cm})^2 = 6 \times 4.00 \, \text{cm}^2 = 24.00 \, \text{cm}^2 \]

4. **Calculate the uncertainty in volume (\( \Delta V \)):**
- The formula for the percentage error in volume is:
\[ \frac{\Delta V}{V} = 3 \times \frac{\Delta A}{A} \]
- Substituting the values:
\[ \frac{\Delta V}{8.00} = 3 \times \frac{0.01}{2.00} \]
- Simplifying:
\[ \frac{\Delta V}{8.00} = 3 \times 0.005 = 0.015 \]
- Therefore:
\[ \Delta V = 0.015 \times 8.00 = 0.12 \, \text{cm}^3 \]

5. **Calculate the uncertainty in surface area (\( \Delta S \)):**
- The formula for the percentage error in surface area is:
\[ \frac{\Delta S}{S} = 2 \times \frac{\Delta A}{A} \]
- Substituting the values:
\[ \frac{\Delta S}{24.00} = 2 \times \frac{0.01}{2.00} \]
- Simplifying:
\[ \frac{\Delta S}{24.00} = 2 \times 0.005 = 0.01 \]
- Therefore:
\[ \Delta S = 0.01 \times 24.00 = 0.24 \, \text{cm}^2 \]

6. **Final results:**
- Volume of the cube with uncertainty:
\[ V = 8.00 \pm 0.12 \, \text{cm}^3 \]
- Surface area of the cube with uncertainty:
\[ S = 24.00 \pm 0.24 \, \text{cm}^2 \]

### Conclusion:
The volume and surface area of the cube are:
- Volume: \( 8.00 \pm 0.12 \, \text{cm}^3 \)
- Surface Area: \( 24.00 \pm 0.24 \, \text{cm}^2 \)