Question

The number of cubes of each side 1 nm and the collective surface area that can be carved out from a cube with each side of 1 m are respectively:

• A. \( 1 \times 10^{27} \) and 6000 km²
• B. \( 1 \times 10^{27} \) and 600 km²
• C. \( 1 \times 10^{25} \) and 5000 km²
• D. \( 1 \times 10^{27} \) and 500 km²

Hint:

When converting from nm to m and m² to km², remember the conversion factors: \( 1 \, \text{m} = 10^9 \, \text{nm} \) and \( 1 \, \text{m}^2 = 10^{-6} \, \text{km}^2 \).

Answer 1

The Correct Option is B

Step 1: Understanding the dimensions.
Given that the side length of the cube is 1 m, we can calculate the number of smaller cubes (each side 1 nm) that can be carved from it.
Since \( 1 \, \text{m} = 10^9 \, \text{nm} \), the volume of the large cube is \( (10^9)^3 \, \text{nm}^3 = 10^{27} \, \text{nm}^3 \).
Thus, the number of cubes with side 1 nm is \( 10^{27} \).

Step 2: Calculating the collective surface area.
The surface area of each small cube is \( 6 \times (1 \, \text{nm})^2 = 6 \, \text{nm}^2 \).
For the entire collection of cubes, the total surface area is \( 6 \times 10^{27} \, \text{nm}^2 \).
Converting this to km²: \[ 6 \times 10^{27} \, \text{nm}^2 = 6 \times 10^{27} \times 10^{-18} \, \text{m}^2 = 6 \times 10^9 \, \text{m}^2 = 6000 \, \text{km}^2. \]
Final Answer: \[ \boxed{(2) \, 1 \times 10^{27} \, \text{and} \, 6000 \, \text{km}^2} \]