How many cm³ are there in 2.5 m³? 2.5 x 102 2.5 x 10-6 2.5 x 106 2.5 x 10-2 none of the above
How many cm³ are there in 2.5 m³? 2.5 x 102 2.5 x 10-6 2.5 x 106 2.5 x 10-2 none of the above
The Correct Answer and Explanation is:
The correct answer is 2.5 x 10⁶.
Explanation:
This problem requires converting a unit of volume, specifically from cubic meters (m³) to cubic centimeters (cm³). To solve this, we must first understand the fundamental relationship between meters (m) and centimeters (cm) and then apply it to three-dimensional space.
1. Establish the Linear Relationship:
The base unit of length in this problem is the meter. The prefix “centi-” means one-hundredth (1/100 or 10⁻²). This gives us the primary conversion factor for length:
- 1 meter = 100 centimeters
In scientific notation, this is expressed as:
- 1 m = 10² cm
2. Convert to Volumetric Relationship:
The question deals with volume, which is a three-dimensional measurement (length × width × height). Therefore, we cannot use the linear conversion factor directly. We must cube the entire linear relationship to find the volumetric equivalent.
Imagine a cube with each side measuring 1 meter. Its volume is 1 m × 1 m × 1 m = 1 m³. To find this volume in cubic centimeters, we convert each side’s length to centimeters first:
- Volume = (100 cm) × (100 cm) × (100 cm)
- Volume = (100)³ cm³
- Volume = 1,000,000 cm³
Using scientific notation, which is more convenient for this calculation:
- 1 m³ = (10² cm)³
- 1 m³ = 10²ˣ³ cm³
- 1 m³ = 10⁶ cm³
So, one cubic meter is equal to one million cubic centimeters. This is our conversion factor for volume.
3. Apply the Conversion Factor:
Now, we can convert the given value of 2.5 m³ to cm³ by multiplying it by our derived conversion factor:
- 2.5 m³ × (10⁶ cm³ / 1 m³)
The m³ units cancel out, leaving us with the answer in cm³:
- 2.5 × 10⁶ cm³
Comparing this result to the given options, we find that it matches the third choice.
