A cuboid of sides 5 cm, 10 cm and 20 cm are melted to form a new cube. What is the ratio between the total surface area of the cuboid and that of the cube ?

Options

A. `6:5`
B. `7:6`
C. `11:10`
D. `9:7`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
Given the dimensions of the cuboid are:
- Length = 5 cm
- Breadth = 10 cm
- Height = 20 cm

Substituting the values:
\[ V = 5 \, \text{cm} \times 10 \, \text{cm} \times 20 \, \text{cm} = 1000 \, \text{cm}^3 \]

### Step 2: Volume of the Cube
Since the cuboid is melted to form a cube, the volume of the cube will be equal to the volume of the cuboid:
\[ V_{\text{cube}} = V_{\text{cuboid}} = 1000 \, \text{cm}^3 \]

### Step 3: Calculate the Side of the Cube
The volume \( V \) of a cube is given by:
\[ V = \text{side}^3 \]
Let the side of the cube be \( A \). Therefore:
\[ A^3 = 1000 \]
Taking the cube root:
\[ A = \sqrt[3]{1000} = 10 \, \text{cm} \]

### Step 4: Calculate the Surface Area of the Cuboid
The surface area \( SA \) of a cuboid is given by the formula:
\[ SA = 2(\text{length} \times \text{breadth} + \text{breadth} \times \text{height} + \text{height} \times \text{length}) \]
Substituting the dimensions:
\[ SA = 2(5 \times 10 + 10 \times 20 + 20 \times 5) \]
Calculating each term:
- \( 5 \times 10 = 50 \)
- \( 10 \times 20 = 200 \)
- \( 20 \times 5 = 100 \)

Thus:
\[ SA = 2(50 + 200 + 100) = 2(350) = 700 \, \text{cm}^2 \]

### Step 5: Calculate the Surface Area of the Cube
The surface area \( SA \) of a cube is given by:
\[ SA = 6 \times \text{side}^2 \]
Substituting the value of \( A \):
\[ SA = 6 \times (10)^2 = 6 \times 100 = 600 \, \text{cm}^2 \]

### Step 6: Calculate the Ratio of Surface Areas
Now, we find the ratio of the surface area of the cuboid to the surface area of the cube:
\[ \text{Ratio} = \frac{SA_{\text{cuboid}}}{SA_{\text{cube}}} = \frac{700}{600} = \frac{7}{6} \]

### Conclusion
The ratio between the total surface area of the cuboid and that of the cube is:
\[ \boxed{\frac{7}{6}} \]