To determine the ranking of the objects from most to least dense, we will first calculate the densities of each object using the formula for density:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
Here are the given values:
- Object [tex]\( W \)[/tex]: Mass [tex]\( = 16 \, \text{g} \)[/tex], Volume [tex]\( = 84 \, \text{cm}^3 \)[/tex]
- Object [tex]\( X \)[/tex]: Mass [tex]\( = 12 \, \text{g} \)[/tex], Volume [tex]\( = 5 \, \text{cm}^3 \)[/tex]
- Object [tex]\( Y \)[/tex]: Mass [tex]\( = 4 \, \text{g} \)[/tex], Volume [tex]\( = 6 \, \text{cm}^3 \)[/tex]
- Object [tex]\( Z \)[/tex]: Mass [tex]\( = 408 \, \text{g} \)[/tex], Volume [tex]\( = 216 \, \text{cm}^3 \)[/tex]
Let's calculate the density for each object:
1. Density of [tex]\( W \)[/tex]:
[tex]\[ \text{Density}_W = \frac{16 \, \text{g}}{84 \, \text{cm}^3} \approx 0.190476 \, \text{g/cm}^3 \][/tex]
2. Density of [tex]\( X \)[/tex]:
[tex]\[ \text{Density}_X = \frac{12 \, \text{g}}{5 \, \text{cm}^3} = 2.4 \, \text{g/cm}^3 \][/tex]
3. Density of [tex]\( Y \)[/tex]:
[tex]\[ \text{Density}_Y = \frac{4 \, \text{g}}{6 \, \text{cm}^3} \approx 0.666667 \, \text{g/cm}^3 \][/tex]
4. Density of [tex]\( Z \)[/tex]:
[tex]\[ \text{Density}_Z = \frac{408 \, \text{g}}{216 \, \text{cm}^3} \approx 1.888889 \, \text{g/cm}^3 \][/tex]
Now, let's rank the objects from most to least dense:
- [tex]\( X \)[/tex]: [tex]\( 2.4 \, \text{g/cm}^3 \)[/tex]
- [tex]\( Z \)[/tex]: [tex]\( \approx 1.888889 \, \text{g/cm}^3 \)[/tex]
- [tex]\( Y \)[/tex]: [tex]\( \approx 0.666667 \, \text{g/cm}^3 \)[/tex]
- [tex]\( W \)[/tex]: [tex]\( \approx 0.190476 \, \text{g/cm}^3 \)[/tex]
Thus, the correct ranking of objects from most to least dense is:
[tex]\[ X, Z, Y, W \][/tex]
So, the correct answer is:
[tex]\[ \boxed{X, Z, Y, W} \][/tex]
