The length breadth and heigth of a cuboid (rectangular solid ) are 4: 3: 2
(i) If its surface area is ` 2548 cm ^(2)` find its volume.
(ii) If its volume is ` 3000 m ^(3),` find its surface area.

Answer

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (i): Finding the Volume of the Cuboid

1. **Define the Dimensions of the Cuboid**:
- Let the length (l) be \(4x\), the breadth (b) be \(3x\), and the height (h) be \(2x\).

2. **Use the Surface Area Formula**:
- The formula for the total surface area (SA) of a cuboid is:
\[ SA = 2(lb + bh + hl) \]
- Substitute the values of l, b, and h:
\[ SA = 2(4x \cdot 3x + 3x \cdot 2x + 2x \cdot 4x) \]
- This simplifies to:
\[ SA = 2(12x^2 + 6x^2 + 8x^2) = 2(26x^2) = 52x^2 \]

3. **Set the Surface Area Equal to 2548 cm²**:
- We know the surface area is 2548 cm²:
\[ 52x^2 = 2548 \]

4. **Solve for \(x^2\)**:
- Divide both sides by 52:
\[ x^2 = \frac{2548}{52} = 49 \]

5. **Find \(x\)**:
- Take the square root of both sides:
\[ x = 7 \text{ cm} \]

6. **Calculate Length, Breadth, and Height**:
- Length \(l = 4x = 4 \cdot 7 = 28 \text{ cm}\)
- Breadth \(b = 3x = 3 \cdot 7 = 21 \text{ cm}\)
- Height \(h = 2x = 2 \cdot 7 = 14 \text{ cm}\)

7. **Calculate the Volume**:
- The formula for the volume (V) of a cuboid is:
\[ V = l \cdot b \cdot h \]
- Substitute the values:
\[ V = 28 \cdot 21 \cdot 14 \]
- Calculate:
\[ V = 8232 \text{ cm}^3 \]

### Part (ii): Finding the Surface Area Given the Volume

1. **Given Volume**:
- The volume is given as 3000 m³.

2. **Set Up the Volume Equation**:
- The volume formula is:
\[ V = l \cdot b \cdot h = 3000 \]
- Substitute \(l = 4x\), \(b = 3x\), \(h = 2x\):
\[ 4x \cdot 3x \cdot 2x = 3000 \]
- This simplifies to:
\[ 24x^3 = 3000 \]

3. **Solve for \(x^3\)**:
- Divide both sides by 24:
\[ x^3 = \frac{3000}{24} = 125 \]

4. **Find \(x\)**:
- Take the cube root of both sides:
\[ x = 5 \text{ m} \]

5. **Calculate Length, Breadth, and Height**:
- Length \(l = 4x = 4 \cdot 5 = 20 \text{ m}\)
- Breadth \(b = 3x = 3 \cdot 5 = 15 \text{ m}\)
- Height \(h = 2x = 2 \cdot 5 = 10 \text{ m}\)

6. **Calculate the Surface Area**:
- Use the surface area formula:
\[ SA = 2(lb + bh + hl) \]
- Substitute the values:
\[ SA = 2(20 \cdot 15 + 15 \cdot 10 + 10 \cdot 20) \]
- Calculate:
\[ SA = 2(300 + 150 + 200) = 2(650) = 1300 \text{ m}^2 \]

### Final Answers:
- (i) The volume of the cuboid is **8232 cm³**.
- (ii) The surface area of the cuboid is **1300 m²**.