A cuboid of sides 9 cm, 27 cm and 24 cm is melted to form a cube. What is the ratio between the total surface area of the cuboid and that of the cube ?

Options

A. `41:35`
B. `41:36`
C. `47:38`
D. `49:36`

Answer

To solve the problem step by step, we need to find the total surface area of the cuboid and the cube formed from it, and then calculate the ratio of these two areas.

### 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 given cuboid with dimensions 9 cm, 27 cm, and 24 cm:
\[ V = 9 \, \text{cm} \times 27 \, \text{cm} \times 24 \, \text{cm} \]
Calculating this:
\[ V = 9 \times 27 = 243 \, \text{cm}^2 \]
\[ V = 243 \times 24 = 5832 \, \text{cm}^3 \]

### Step 2: Determine the Side Length 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. Let the side length of the cube be \( a \). The volume \( V \) of a cube is given by:
\[ V = a^3 \]
Setting the volumes equal:
\[ a^3 = 5832 \]
To find \( a \), we take the cube root:
\[ a = \sqrt[3]{5832} \]
Calculating this gives:
\[ a = 18 \, \text{cm} \]

### 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 dimensions:
\[ A = 2(9 \times 27 + 27 \times 24 + 24 \times 9) \]
Calculating each term:
\[ 9 \times 27 = 243 \]
\[ 27 \times 24 = 648 \]
\[ 24 \times 9 = 216 \]
Now substituting back into the area formula:
\[ A = 2(243 + 648 + 216) = 2(1107) = 2214 \, \text{cm}^2 \]

### 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 \( a = 18 \, \text{cm} \):
\[ A = 6 \times (18)^2 = 6 \times 324 = 1944 \, \text{cm}^2 \]

### Step 5: Calculate the Ratio of the Total Surface Areas
Now, we find the ratio of the total surface area of the cuboid to that of the cube:
\[ \text{Ratio} = \frac{\text{Surface Area of Cuboid}}{\text{Surface Area of Cube}} = \frac{2214}{1944} \]
To simplify this ratio, we can divide both numbers by their greatest common divisor (GCD). After simplifying:
\[ \text{Ratio} = \frac{2214 \div 18}{1944 \div 18} = \frac{123}{108} = \frac{41}{36} \]

### Final Answer
The ratio between the total surface area of the cuboid and that of the cube is:
\[ \frac{41}{36} \]