Find the volume, lateral surface area and the total surface area of a cube each of whose edges measures (i) 7m (ii) 5.6 cm (iii) 8 dm, 5 cm

Answer

To solve the problem of finding the volume, lateral surface area, and total surface area of a cube for given edge lengths, we will follow these steps for each case.

### Given:
- **Edge length (a)** for each case:
1. \( a = 7 \, \text{m} \)
2. \( a = 5.6 \, \text{cm} \)
3. \( a = 8 \, \text{dm} + 5 \, \text{cm} \)

### Formulas:
1. **Volume of a cube**: \( V = a^3 \)
2. **Lateral Surface Area (LSA)**: \( LSA = 4a^2 \)
3. **Total Surface Area (TSA)**: \( TSA = 6a^2 \)

### Step-by-Step Solutions:

#### (i) For \( a = 7 \, \text{m} \):
1. **Volume**:
\[ V = a^3 = 7^3 = 7 \times 7 \times 7 = 343 \, \text{m}^3 \]

2. **Lateral Surface Area**:
\[ LSA = 4a^2 = 4 \times 7^2 = 4 \times 49 = 196 \, \text{m}^2 \]

3. **Total Surface Area**:
\[ TSA = 6a^2 = 6 \times 7^2 = 6 \times 49 = 294 \, \text{m}^2 \]

#### (ii) For \( a = 5.6 \, \text{cm} \):
1. **Volume**:
\[ V = a^3 = 5.6^3 = 5.6 \times 5.6 \times 5.6 = 175.616 \, \text{cm}^3 \]

2. **Lateral Surface Area**:
\[ LSA = 4a^2 = 4 \times 5.6^2 = 4 \times 31.36 = 125.44 \, \text{cm}^2 \]

3. **Total Surface Area**:
\[ TSA = 6a^2 = 6 \times 5.6^2 = 6 \times 31.36 = 188.16 \, \text{cm}^2 \]

#### (iii) For \( a = 8 \, \text{dm} + 5 \, \text{cm} \):
1. **Convert dimensions**:
- \( 8 \, \text{dm} = 80 \, \text{cm} \)
- Therefore, \( a = 80 \, \text{cm} + 5 \, \text{cm} = 85 \, \text{cm} \)

2. **Volume**:
\[ V = a^3 = 85^3 = 85 \times 85 \times 85 = 614125 \, \text{cm}^3 \]

3. **Lateral Surface Area**:
\[ LSA = 4a^2 = 4 \times 85^2 = 4 \times 7225 = 28900 \, \text{cm}^2 \]

4. **Total Surface Area**:
\[ TSA = 6a^2 = 6 \times 85^2 = 6 \times 7225 = 43350 \, \text{cm}^2 \]

### Summary of Results:
1. For \( a = 7 \, \text{m} \):
- Volume: \( 343 \, \text{m}^3 \)
- Lateral Surface Area: \( 196 \, \text{m}^2 \)
- Total Surface Area: \( 294 \, \text{m}^2 \)

2. For \( a = 5.6 \, \text{cm} \):
- Volume: \( 175.616 \, \text{cm}^3 \)
- Lateral Surface Area: \( 125.44 \, \text{cm}^2 \)
- Total Surface Area: \( 188.16 \, \text{cm}^2 \)

3. For \( a = 8 \, \text{dm} + 5 \, \text{cm} \):
- Volume: \( 614125 \, \text{cm}^3 \)
- Lateral Surface Area: \( 28900 \, \text{cm}^2 \)
- Total Surface Area: \( 43350 \, \text{cm}^2 \)