A cuboid which sides 6 cm, 9 cm and 32 cm is 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. `93:71`
B. `108:113`
C. `297:220`
D. `89:72`

Answer

To solve the problem step by step, we will first calculate the volume of the cuboid, then find the side length of the cube formed from that volume, and finally compute the total surface areas of both the cuboid and the cube to find their ratio.

### Step 1: Calculate the Volume of the Cuboid
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
For the cuboid with dimensions 6 cm, 9 cm, and 32 cm:
\[ V = 6 \times 9 \times 32 \]
Calculating this:
\[ V = 54 \times 32 = 1728 \text{ cm}^3 \]

### Step 2: Determine the Side Length of the Cube
Since the volume of the cube is equal to the volume of the cuboid, we have:
\[ \text{Volume of the cube} = 1728 \text{ cm}^3 \]
Let \( a \) be the side length of the cube. The volume of the cube is given by:
\[ V = a^3 \]
Setting the volumes equal:
\[ a^3 = 1728 \]
To find \( a \), we take the cube root:
\[ a = \sqrt[3]{1728} = 12 \text{ cm} \]

### Step 3: Calculate the Total Surface Area of the Cuboid
The total surface area \( A \) of a cuboid is given by the formula:
\[ A = 2(lb + bh + hl) \]
Substituting the values:
\[ A = 2(6 \times 9 + 9 \times 32 + 32 \times 6) \]
Calculating each term:
- \( 6 \times 9 = 54 \)
- \( 9 \times 32 = 288 \)
- \( 32 \times 6 = 192 \)

Now substituting back:
\[ A = 2(54 + 288 + 192) = 2(534) = 1068 \text{ cm}^2 \]

### Step 4: Calculate the Total Surface Area of the Cube
The total surface area \( A \) of a cube is given by:
\[ A = 6a^2 \]
Substituting \( a = 12 \text{ cm} \):
\[ A = 6 \times (12)^2 = 6 \times 144 = 864 \text{ cm}^2 \]

### Step 5: Calculate the Ratio of the Surface Areas
Now, we find the ratio of the total surface area of the cuboid to that of the cube:
\[ \text{Ratio} = \frac{\text{Surface Area of Cuboid}}{\text{Surface Area of Cube}} = \frac{1068}{864} \]
To simplify this ratio:
\[ \text{Ratio} = \frac{1068 \div 12}{864 \div 12} = \frac{89}{72} \]

### Final Answer
Thus, the ratio between the total surface area of the cuboid and that of the cube is:
\[ \text{Ratio} = 89 : 72 \]