Cube Surface Area Ratio Calculation

Let's find the ratio of the surface area of the original cube to the total surface area of all the smaller cubes formed.

Understanding the Cubes and Dimensions

• Original cube side length: $s_{large} = 1$ cm
• Smaller cube side length: $s_{small} = 1$ mm

To compare and calculate volumes and areas, we need to use the same unit. Let's convert centimeters to millimeters:

$1$ cm $= 10$ mm

So, the side length of the original cube is $s_{large} = 10$ mm.

Calculating Surface Area of the Original Cube

The surface area of a cube with side length $s$ is given by $6s^2$ (since there are 6 faces, each with area $s^2$).

Surface area of the large cube ($SA_{large}$):

$SA_{large} = 6 \times (s_{large})^2$

$SA_{large} = 6 \times (10 \text{ mm})^2$

$SA_{large} = 6 \times 100 \text{ mm}^2$

$SA_{large} = 600 \text{ mm}^2$

Determining the Number of Smaller Cubes

When the large cube is broken into smaller identical cubes, the total volume remains the same. We can find the number of small cubes by dividing the volume of the large cube by the volume of one small cube.

Volume of a cube with side length $s$ is given by $s^3$.

Volume of the large cube ($V_{large}$):

$V_{large} = (s_{large})^3 = (10 \text{ mm})^3 = 1000 \text{ mm}^3$

Volume of one small cube ($V_{small}$):

$V_{small} = (s_{small})^3 = (1 \text{ mm})^3 = 1 \text{ mm}^3$

Number of small cubes ($N$):

$N = \frac{V_{large}}{V_{small}} = \frac{1000 \text{ mm}^3}{1 \text{ mm}^3} = 1000$

So, 1000 smaller cubes of side 1 mm are formed from the original 1 cm cube.

Calculating Total Surface Area of Smaller Cubes

First, calculate the surface area of one small cube:

$SA_{small} = 6 \times (s_{small})^2$

$SA_{small} = 6 \times (1 \text{ mm})^2$

$SA_{small} = 6 \times 1 \text{ mm}^2$

$SA_{small} = 6 \text{ mm}^2$

Now, calculate the total surface area of all 1000 smaller cubes:

$SA_{total\_small} = N \times SA_{small}$

$SA_{total\_small} = 1000 \times 6 \text{ mm}^2$

$SA_{total\_small} = 6000 \text{ mm}^2$

Finding the Ratio of Surface Areas

The question asks for the ratio of the surface area of the original cube to the total surface area of the smaller cubes.

Ratio $= \frac{SA_{large}}{SA_{total\_small}}$

Ratio $= \frac{600 \text{ mm}^2}{6000 \text{ mm}^2}$

Ratio $= \frac{600}{6000}$

Ratio $= \frac{6}{60}$

Ratio $= \frac{1}{10}$

The ratio of the surface area of the original cube to the total surface area of the smaller cubes is 1/10.

Summary of Calculations

Ratio $= \frac{\text{Surface Area of Original Cube}}{\text{Total Surface Area of Small Cubes}} = \frac{600 \text{ mm}^2}{6000 \text{ mm}^2} = \frac{1}{10}$