The volume of a cube is four times the volume of a cuboid. If the sides of the cuboid are 32 cm, 8 cm and 4 cm, then find the ratio of the total surface area of the cube to that of the cuboid.

The correct answer is Option 2: 24 : 13.

Step 1: Calculate the volume of the cuboid
The dimensions of the cuboid are given as:
Length ($l$) = 32 cm
Breadth ($b$) = 8 cm
Height ($h$) = 4 cm
The volume of a cuboid is calculated using the formula: Volume = $l\times b\times h$.

$\text{Volume of cuboid}=32\times 8\times 4=1024{\text{cm}}^3$

Step 2: Calculate the volume and side length of the cube
It is given that the volume of the cube is four times the volume of the cuboid.

$\text{Volume of cube}=4\times 1024=4096{\text{cm}}^3$

Let the side of the cube be $a$. The volume of a cube is given by $a^3$.

$a^3=4096$

Taking the cube root on both sides:

$a=\sqrt[3]{4096}=16\text{cm}$

Step 3: Calculate the total surface area of the cube
The total surface area of a cube is given by the formula: $6a^2$.

$\text{Total surface area of cube}=6\times {\left(16\right)}^2=6\times 256=1536{\text{cm}}^2$

Step 4: Calculate the total surface area of the cuboid
The total surface area of a cuboid is given by the formula: $2(lb+bh+hl)$.

$\text{Total surface area of cuboid}=2\times (32\times 8+8\times 4+4\times 32)$

$=2\times (256+32+128)$

$=2\times 416=832{\text{cm}}^2$

Step 5: Find the required ratio
We need to find the ratio of the total surface area of the cube to that of the cuboid.

$\text{Ratio}=\frac{1536}{832}$

Dividing both numerator and denominator by their common factor (64):

$\text{Ratio}=\frac{24}{13}$

Thus, the ratio of the total surface area of the cube to that of the cuboid is 24 : 13.