To convert the density from pounds per cubic foot (lb/ft³) to kilograms per cubic meter (kg/m³), you need to use two conversion ratios: one for mass and one for volume. The relevant ratios are:
- [tex]\(\frac{1 \text{ kg}}{2.2 \text{ lb}}\)[/tex]
- [tex]\(\frac{1 \text{ m}^3}{35.3 \text{ ft}^3}\)[/tex]
Given that the density is 167 lb/ft³, let's place these in the equation to get the density in kg/m³:
[tex]\[
167 \frac{\text{lb}}{\text{ft}^3} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} \times \frac{1 \text{m}^3}{35.3 \text{ft}^3} = ? \frac{\text{kg}}{\text{m}^3}
\][/tex]
To solve this:
1. Multiply 167 by [tex]\(\frac{1 \text{ kg}}{2.2 \text{ lb}}\)[/tex]:
[tex]\[
167 \times \frac{1 \text{ kg}}{2.2 \text{ lb}} = 75.9090909090909 \frac{\text{kg}}{\text{ft}^3}
\][/tex]
2. Now, take the result and multiply by [tex]\(\frac{1 \text{ m}^3}{35.3 \text{ ft}^3}\)[/tex]:
[tex]\[
75.9090909090909 \times \frac{1 \text{ m}^3}{35.3 \text{ ft}^3} = 2.15039917589493 \frac{\text{kg}}{\text{m}^3}
\][/tex]
Thus, the crate's density in kilograms per cubic meter is:
[tex]\[
2.15039917589493 \frac{\text{kg}}{\text{m}^3}
\][/tex]
Rounded to the nearest hundredth, the density is:
[tex]\[
2.15 \frac{\text{kg}}{\text{m}^3}
\][/tex]
Therefore, the final answer, rounded to the nearest hundredth, is:
[tex]\[
2.15 \frac{\text{kg}}{\text{m}^3}
\][/tex]
