Cube Surface Area Ratio Problem

This problem asks us to find the ratio of the surface area of a large cube to the total surface area of many smaller identical cubes formed by breaking the larger one. We are given the side lengths of the large and small cubes.

Understanding the Dimensions

We have a large cube with a side length of 1 cm and it is broken into smaller identical cubes, each with a side length of 1 mm.

• Side of the large cube ($S_{large}$) = 1 cm
• Side of the small cube ($s_{small}$) = 1 mm

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

$1 \text{ cm} = 10 \text{ mm}$

So, the side of the large cube is 10 mm.

Calculating the Number of Small Cubes

When a large cube is broken into smaller identical cubes, the total volume remains the same. First, let's calculate the volume of the large cube and a single small cube.

• Volume of the large cube ($V_{large}$) = $(S_{large})^3 = (10 \text{ mm})^3 = 10 \text{ mm} \times 10 \text{ mm} \times 10 \text{ mm} = 1000 \text{ mm}^3$.
• Volume of a small cube ($v_{small}$) = $(s_{small})^3 = (1 \text{ mm})^3 = 1 \text{ mm} \times 1 \text{ mm} \times 1 \text{ mm} = 1 \text{ mm}^3$.

The number of small cubes ($N$) that can be formed from the large cube is the ratio of their volumes:

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

So, there are 1000 small cubes formed.

Surface Area Calculations

The surface area of a cube with side 's' is given by the formula $6s^2$, because a cube has 6 identical square faces.

• Surface area of the large cube ($A_{large}$) = $6 \times (S_{large})^2 = 6 \times (10 \text{ mm})^2 = 6 \times 100 \text{ mm}^2 = 600 \text{ mm}^2$.
• Surface area of one small cube ($a_{small}$) = $6 \times (s_{small})^2 = 6 \times (1 \text{ mm})^2 = 6 \times 1 \text{ mm}^2 = 6 \text{ mm}^2$.

The problem asks for the total surface area of all the small cubes. Since there are 1000 small cubes, the total surface area is:

Total surface area of small cubes ($A_{total\_small}$) = $N \times a_{small} = 1000 \times 6 \text{ mm}^2 = 6000 \text{ mm}^2$.

Calculating the Ratio

We need to find the ratio of the surface area of the large cube to the total surface area of the small cubes.

Ratio = $\frac{\text{Surface area of large cube}}{\text{Total surface area of small cubes}} = \frac{A_{large}}{A_{total\_small}}$

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

Simplifying the fraction:

Ratio = $\frac{6}{60} = \frac{1}{10}$

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