To understand why [tex]\(1 \, \text{yd}^3\)[/tex] is equal to [tex]\(27 \, \text{ft}^3\)[/tex], let's break the problem down step-by-step.
1. Understanding the Conversion Factor:
- 1 yard (yd) is equivalent to 3 feet (ft). This is a linear measurement.
2. Volume Conversion:
- When converting volume, we need to consider three dimensions (length, width, and height).
3. Step-by-Step Conversion:
- To convert from cubic yards to cubic feet, we need to convert each dimension from yards to feet.
- Since 1 yard is 3 feet, we need to cube this conversion because volume is a function of all three dimensions, i.e., length, width, and height.
4. Mathematical Representation:
- Start with the volume in cubic yards: [tex]\(1 \, \text{yd}^3\)[/tex].
- We know that [tex]\(1 \, \text{yd} \)[/tex] is [tex]\(3 \, \text{ft}\)[/tex].
- Therefore, [tex]\(1 \, \text{yd}^3 = (1 \, \text{yd}) \times (1 \, \text{yd}) \times (1 \, \text{yd})\)[/tex].
- Substituting yards with feet: [tex]\(1 \, \text{yd}^3 = (3 \, \text{ft}) \times (3 \, \text{ft}) \times (3 \, \text{ft})\)[/tex].
5. Simplifying the Product:
- Multiply the values together: [tex]\( (3 \, \text{ft}) \times (3 \, \text{ft}) \times (3 \, \text{ft}) = 27 \, \text{ft}^3 \)[/tex].
Hence, [tex]\(1 \, \text{yd}^3 = 27 \, \text{ft}^3\)[/tex]. This clearly shows that [tex]\(1 \, \text{yd}^3 \ne 3 \, \text{ft}^3\)[/tex]; instead, it equals [tex]\(27 \, \text{ft}^3\)[/tex] because the volume conversion must account for all three dimensions.
