This table shows the mass and volume of four different objects:

\begin{tabular}{|l|l|}
\hline
Object & Measurements \\
\hline
W & \begin{tabular}{l}
Mass: [tex][tex]$16 \, \text{g}$[/tex][/tex] \\
Volume: [tex][tex]$84 \, \text{cm}^3$[/tex][/tex]
\end{tabular} \\
\hline
[tex][tex]$X$[/tex][/tex] & \begin{tabular}{l}
Mass: [tex][tex]$12 \, \text{g}$[/tex][/tex] \\
Volume: [tex][tex]$5 \, \text{cm}^3$[/tex][/tex]
\end{tabular} \\
\hline
[tex][tex]$Y$[/tex][/tex] & \begin{tabular}{l}
Mass: [tex][tex]$4 \, \text{g}$[/tex][/tex] \\
Volume: [tex][tex]$6 \, \text{cm}^3$[/tex][/tex]
\end{tabular} \\
\hline
[tex][tex]$Z$[/tex][/tex] & \begin{tabular}{l}
Mass: [tex][tex]$408 \, \text{g}$[/tex][/tex] \\
Volume: [tex][tex]$216 \, \text{cm}^3$[/tex][/tex]
\end{tabular} \\
\hline
\end{tabular}

Which ranks the objects from most to least dense?

A. [tex][tex]$X, Y, W, Z$[/tex][/tex]
B. [tex][tex]$X, Z, Y, W$[/tex][/tex]
C. [tex][tex]$W, Y, Z, X$[/tex][/tex]
D. [tex][tex]$Z, Y, X, W$[/tex][/tex]



Answer :

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]