Find the volume, lateral surface area and the total surface area of the cuboid whose dimensions are
(i) length =22 cm, breadth = 12 cm and height =7.5 cm
(ii) length =15 m, breadth =6 m and height =9dm
(iii) lenth =24m, breadth =25 cm and height =6m
(iv) length =48 cm, breadth =6 dm and height =1 m

Answer

To find the volume, lateral surface area, and total surface area of a cuboid, we will use the following formulas:

1. **Volume (V)** = Length (L) × Breadth (B) × Height (H)
2. **Lateral Surface Area (LSA)** = 2 × Height (H) × (Length (L) + Breadth (B))
3. **Total Surface Area (TSA)** = 2 × (Length (L) × Breadth (B) + Breadth (B) × Height (H) + Height (H) × Length (L))

Now, let's solve the given parts step by step.

### (i) Length = 22 cm, Breadth = 12 cm, Height = 7.5 cm

**Step 1: Calculate the Volume**
\[ V = L \times B \times H = 22 \, \text{cm} \times 12 \, \text{cm} \times 7.5 \, \text{cm} = 1980 \, \text{cm}^3 \]

**Step 2: Calculate the Lateral Surface Area**
\[ LSA = 2 \times H \times (L + B) = 2 \times 7.5 \, \text{cm} \times (22 \, \text{cm} + 12 \, \text{cm}) = 2 \times 7.5 \times 34 = 510 \, \text{cm}^2 \]

**Step 3: Calculate the Total Surface Area**
\[ TSA = 2 \times (L \times B + B \times H + H \times L) = 2 \times (22 \times 12 + 12 \times 7.5 + 7.5 \times 22) \]
Calculating each term:
- \(22 \times 12 = 264\)
- \(12 \times 7.5 = 90\)
- \(7.5 \times 22 = 165\)

Now, substituting back:
\[ TSA = 2 \times (264 + 90 + 165) = 2 \times 519 = 1038 \, \text{cm}^2 \]

### (ii) Length = 15 m, Breadth = 6 m, Height = 9 dm

**Step 1: Convert Height to Meters**
\[ 9 \, \text{dm} = 0.9 \, \text{m} \]

**Step 2: Calculate the Volume**
\[ V = L \times B \times H = 15 \, \text{m} \times 6 \, \text{m} \times 0.9 \, \text{m} = 81 \, \text{m}^3 \]

**Step 3: Calculate the Lateral Surface Area**
\[ LSA = 2 \times H \times (L + B) = 2 \times 0.9 \, \text{m} \times (15 \, \text{m} + 6 \, \text{m}) = 2 \times 0.9 \times 21 = 37.8 \, \text{m}^2 \]

**Step 4: Calculate the Total Surface Area**
\[ TSA = 2 \times (L \times B + B \times H + H \times L) = 2 \times (15 \times 6 + 6 \times 0.9 + 0.9 \times 15) \]
Calculating each term:
- \(15 \times 6 = 90\)
- \(6 \times 0.9 = 5.4\)
- \(0.9 \times 15 = 13.5\)

Now, substituting back:
\[ TSA = 2 \times (90 + 5.4 + 13.5) = 2 \times 108.9 = 217.8 \, \text{m}^2 \]

### (iii) Length = 24 m, Breadth = 25 cm, Height = 6 m

**Step 1: Convert Breadth to Meters**
\[ 25 \, \text{cm} = 0.25 \, \text{m} \]

**Step 2: Calculate the Volume**
\[ V = L \times B \times H = 24 \, \text{m} \times 0.25 \, \text{m} \times 6 \, \text{m} = 36 \, \text{m}^3 \]

**Step 3: Calculate the Lateral Surface Area**
\[ LSA = 2 \times H \times (L + B) = 2 \times 6 \, \text{m} \times (24 \, \text{m} + 0.25 \, \text{m}) = 2 \times 6 \times 24.25 = 291 \, \text{m}^2 \]

**Step 4: Calculate the Total Surface Area**
\[ TSA = 2 \times (L \times B + B \times H + H \times L) = 2 \times (24 \times 0.25 + 0.25 \times 6 + 6 \times 24) \]
Calculating each term:
- \(24 \times 0.25 = 6\)
- \(0.25 \times 6 = 1.5\)
- \(6 \times 24 = 144\)

Now, substituting back:
\[ TSA = 2 \times (6 + 1.5 + 144) = 2 \times 151.5 = 303 \, \text{m}^2 \]

### (iv) Length = 48 cm, Breadth = 6 dm, Height = 1 m

**Step 1: Convert all dimensions to meters**
- Length: \(48 \, \text{cm} = 0.48 \, \text{m}\)
- Breadth: \(6 \, \text{dm} = 0.6 \, \text{m}\)
- Height: \(1 \, \text{m} = 1 \, \text{m}\)

**Step 2: Calculate the Volume**
\[ V = L \times B \times H = 0.48 \, \text{m} \times 0.6 \, \text{m} \times 1 \, \text{m} = 0.288 \, \text{m}^3 \]

**Step 3: Calculate the Lateral Surface Area**
\[ LSA = 2 \times H \times (L + B) = 2 \times 1 \, \text{m} \times (0.48 \, \text{m} + 0.6 \, \text{m}) = 2 \times 1 \times 1.08 = 2.16 \, \text{m}^2 \]

**Step 4: Calculate the Total Surface Area**
\[ TSA = 2 \times (L \times B + B \times H + H \times L) = 2 \times (0.48 \times 0.6 + 0.6 \times 1 + 1 \times 0.48) \]
Calculating each term:
- \(0.48 \times 0.6 = 0.288\)
- \(0.6 \times 1 = 0.6\)
- \(1 \times 0.48 = 0.48\)

Now, substituting back:
\[ TSA = 2 \times (0.288 + 0.6 + 0.48) = 2 \times 1.368 = 2.736 \, \text{m}^2 \]

### Summary of Results
1. For dimensions 22 cm, 12 cm, 7.5 cm:
- Volume = 1980 cm³
- Lateral Surface Area = 510 cm²
- Total Surface Area = 1038 cm²

2. For dimensions 15 m, 6 m, 9 dm:
- Volume = 81 m³
- Lateral Surface Area = 37.8 m²
- Total Surface Area = 217.8 m²

3. For dimensions 24 m, 25 cm, 6 m:
- Volume = 36 m³
- Lateral Surface Area = 291 m²
- Total Surface Area = 303 m²

4. For dimensions 48 cm, 6 dm, 1 m:
- Volume = 0.288 m³
- Lateral Surface Area = 2.16 m²
- Total Surface Area = 2.736 m²