Question

On breaking a cubic solid (edge \(=1 \mathrm{~m}\) ) into fine cubic particles of edge \(1 \mu \mathrm{m}\) surface becomes (a) remains same (b) 100 times (c) 1000 times (d) 1000,000 times

Short answer

(d) 1,000,000 times

Step-by-step solution

Understand the Problem

You have a large cube with an edge length of 1 meter. This cube is broken down into smaller cubes with an edge length of 1 micrometer. You need to calculate the change in the total surface area of these cubes.

Calculate Surface Area of Original Cube

The original cube has an edge length of 1 meter. The surface area of a cube with edge length \( a \) is given by \( 6a^2 \), so substitute \( a = 1 \) meter into the formula: \[ 6 \times (1)^2 = 6 \text{ square meters} \]

Calculate Surface Area of a Single Small Cube

Each small cube has an edge of 1 micrometer (\(1 \mu m = 10^{-6} m\)). The surface area of one small cube is \( 6 \times (10^{-6})^2 \): \[ 6 \times (10^{-6})^2 = 6 \times 10^{-12} \text{ square meters} \]

Calculate Number of Small Cubes

The volume of the original cube is \( (1)^3 = 1 \text{ cubic meter} \). The volume of one small cube is \( (10^{-6})^3 = 10^{-18} \text{ cubic meters} \). Divide the original volume by the volume of one small cube to get the number of small cubes:\[ \frac{1}{10^{-18}} = 10^{18} \text{ small cubes} \]

Calculate Total Surface Area of Small Cubes

Multiply the surface area of one small cube by the number of small cubes: \[ 10^{18} \times 6 \times 10^{-12} = 6 \times 10^{6} \text{ square meters} \]

Compare Surface Areas

The surface area of the original cube is 6 square meters, while the total surface area of the small cubes is \(6 \times 10^{6}\) square meters. Divide the large surface area by the original surface area to see how many times larger it is:\[ \frac{6 \times 10^{6}}{6} = 10^{6} \]

Conclusion

The total surface area becomes 1,000,000 times larger after the original cube is broken into smaller cubes.

Key concepts

Cube Geometry

Understanding the geometry of a cube is essential in solving problems involving surface area and volume. A cube is a three-dimensional shape with six square faces, twelve equal edges, and eight vertices. The edge length is crucial, as both the surface area and the volume of a cube depend on it.

When calculating the surface area, you use the formula:

• Surface Area = 6 times the square of edge length, or \(6a^2\).

For volume, the formula is:

• Volume = edge length cubed, or \(a^3\).

Knowing these formulas helps us understand other concepts, like how breaking a cube into smaller parts affects its total surface area.

Unit Conversion

Units of measurement are crucial in solving geometric problems. Being able to convert between units ensures accurate calculations.

For instance, in this exercise, we need to convert between meters and micrometers:

• 1 meter = 1,000,000 micrometers (\(1\,m = 10^6 \mu m\)).
• 1 micrometer = 0.000001 meters (\(1 \mu m = 10^{-6} m\)).

This conversion is necessary when calculating the surface area of the small cubes. By knowing that an edge of 1 micrometer equals \(10^{-6}\) meters, we can easily apply this to our calculations. Remembering unit conversion is a fundamental skill across many areas of mathematics and science.

Multiplicative Scaling

Multiplicative scaling in geometry allows us to understand how dimensions can change proportionally. In the case of a cube, if we reduce the edge length, the surface area and volume change according to specific scaling laws.

When a cube is broken into smaller cubes, each dimension scales down, but the number of units dramatically increases. A meter-sized cube divided into micrometer-sized cubes is an example:

• The edge of the small cubes is \(10^{-6}\) times that of the original, resulting in a vast number of small cubes (\(10^{18}\) small cubes).
• Similarly, the surface area calculation follows: the total surface area becomes multiplied due to the quantity of smaller surfaces, making it \(10^6\) times larger.

This explains why seemingly small alterations to a dimension can result in large-scale changes in properties.

Mathematical Modeling

Mathematical modeling is the practice of representing real-world phenomena with mathematical expressions for easier analysis and visualization.

In this exercise, modeling the cube as it divides into smaller cubes helps visualize the problem and simplify calculations.

• The division of the cube is first represented in terms of edge length, with a large cube broken into innumerable smaller cubes.
• This model allows us to calculate new parameters like the surface area and volume of each cube and the cumulative effect of these changes.Through mathematical modeling, we can predict how the surface area will change and arrive at the conclusion that the surface area increases by a factor of \(1,000,000\).

This kind of modeling educates on how mathematical expressions can simplify investigating and solving complex real-world problems.