A cube of edge 6 cm and a cuboid with dimensions ` 4 cm xx x cm xx 15 cm ` are equals in volume . Find :
(i) the value of x
(ii) total surface area of the cuboid
(iii) total surface area of the cube.
Which of these two has greater surface and by how much?

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Find the value of x

1. **Calculate the volume of the cube:**
The formula for the volume of a cube is:
\[ V = a^3 \]
where \( a \) is the edge length of the cube. Here, \( a = 6 \) cm.
\[ V = 6^3 = 6 \times 6 \times 6 = 216 \text{ cm}^3 \]

2. **Set the volume of the cuboid equal to the volume of the cube:**
The volume of the cuboid is given by:
\[ V = l \times b \times h \]
where \( l = 4 \) cm, \( b = x \) cm, and \( h = 15 \) cm.
\[ V = 4 \times x \times 15 \]

3. **Equate the volumes:**
\[ 216 = 4 \times x \times 15 \]

4. **Simplify the equation:**
\[ 216 = 60x \]

5. **Solve for x:**
\[ x = \frac{216}{60} = 3.6 \text{ cm} \]

### Step 2: Find the total surface area of the cuboid

1. **Use the formula for the total surface area of a cuboid:**
The formula is:
\[ \text{Total Surface Area} = 2(lb + bh + hl) \]

2. **Substitute the values:**
Here, \( l = 4 \) cm, \( b = 3.6 \) cm, and \( h = 15 \) cm.
\[ \text{Total Surface Area} = 2(4 \times 3.6 + 3.6 \times 15 + 15 \times 4) \]

3. **Calculate each term:**
- \( 4 \times 3.6 = 14.4 \)
- \( 3.6 \times 15 = 54 \)
- \( 15 \times 4 = 60 \)

4. **Add the terms:**
\[ 14.4 + 54 + 60 = 128.4 \]

5. **Multiply by 2:**
\[ \text{Total Surface Area} = 2 \times 128.4 = 256.8 \text{ cm}^2 \]

### Step 3: Find the total surface area of the cube

1. **Use the formula for the total surface area of a cube:**
The formula is:
\[ \text{Total Surface Area} = 6a^2 \]

2. **Substitute the value of a:**
Here, \( a = 6 \) cm.
\[ \text{Total Surface Area} = 6 \times 6^2 = 6 \times 36 = 216 \text{ cm}^2 \]

### Step 4: Compare the surface areas

1. **Compare the total surface areas:**
- Total Surface Area of the cuboid = 256.8 cm²
- Total Surface Area of the cube = 216 cm²

2. **Determine which is greater and by how much:**
\[ 256.8 - 216 = 40.8 \text{ cm}^2 \]

### Final Answers:
(i) The value of \( x \) is \( 3.6 \) cm.
(ii) The total surface area of the cuboid is \( 256.8 \) cm².
(iii) The total surface area of the cube is \( 216 \) cm².
The cuboid has a greater surface area by \( 40.8 \) cm².