To determine which ranks the objects from most to least dense, start by calculating the density for each object. Density can be calculated using the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
Given the measurements:
- Object W:
[tex]\[ \text{Mass} = 16 \, \text{g}, \, \text{Volume} = 84 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{16 \, \text{g}}{84 \, \text{cm}^3} \approx 0.1905 \, \text{g/cm}^3 \][/tex]
- Object X:
[tex]\[ \text{Mass} = 12 \, \text{g}, \, \text{Volume} = 5 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{12 \, \text{g}}{5 \, \text{cm}^3} = 2.4 \, \text{g/cm}^3 \][/tex]
- Object Y:
[tex]\[ \text{Mass} = 4 \, \text{g}, \, \text{Volume} = 6 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{4 \, \text{g}}{6 \, \text{cm}^3} \approx 0.6667 \, \text{g/cm}^3 \][/tex]
- Object Z:
[tex]\[ \text{Mass} = 408 \, \text{g}, \, \text{Volume} = 216 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{408 \, \text{g}}{216 \, \text{cm}^3} \approx 1.8889 \, \text{g/cm}^3 \][/tex]
Now, list the densities:
- Object W: [tex]\( 0.1905 \, \text{g/cm}^3 \)[/tex]
- Object X: [tex]\( 2.4 \, \text{g/cm}^3 \)[/tex]
- Object Y: [tex]\( 0.6667 \, \text{g/cm}^3 \)[/tex]
- Object Z: [tex]\( 1.8889 \, \text{g/cm}^3 \)[/tex]
Next, sort the objects by their densities in descending order (from the highest to the lowest):
1. Object X: [tex]\( 2.4 \, \text{g/cm}^3 \)[/tex]
2. Object Z: [tex]\( 1.8889 \, \text{g/cm}^3 \)[/tex]
3. Object Y: [tex]\( 0.6667 \, \text{g/cm}^3 \)[/tex]
4. Object W: [tex]\( 0.1905 \, \text{g/cm}^3 \)[/tex]
Thus, the correct ranking from most to least dense is:
[tex]\[ X, Z, Y, W \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{X, Z, Y, W} \][/tex]
