Researchers collected data to determine volumetric bone density for four samples. The data are recorded in the table.

What is the average bone density for the data samples provided?

A. [tex][tex]$3.2 \, g/cm^3$[/tex][/tex]
B. [tex][tex]$0.36 \, g/cm^3$[/tex][/tex]
C. [tex][tex]$0.31 \, g/cm^3$[/tex][/tex]
D. [tex][tex]$0.03 \, g/cm^3$[/tex][/tex]

Bone Density Table

\begin{tabular}{|c|c|c|}
\hline
Sample & \begin{tabular}{c}
Mass of Sample \\
[tex]$(g)$[/tex]
\end{tabular} & \begin{tabular}{c}
Volume of Sample \\
[tex]$(cm^3)$[/tex]
\end{tabular} \\
\hline
1 & 6.8 & 22.6 \\
\hline
2 & 1.7 & 5.4 \\
\hline
3 & 3.6 & 11.3 \\
\hline
4 & 5.2 & 17.4 \\
\hline
\end{tabular}



Answer :

To calculate the average bone density for the provided data samples, follow these steps:

1. Determine the bone density for each sample:

The formula for bone density is given by:
[tex]\[ \text{Bone Density} = \frac{\text{Mass of Sample}}{\text{Volume of Sample}} \][/tex]

We will apply this formula to each of the provided samples:

- For Sample 1:
[tex]\[ \text{Bone Density}_1 = \frac{6.8 \, \text{g}}{22.6 \, \text{cm}^3} \approx 0.3009 \, \text{g/cm}^3 \][/tex]

- For Sample 2:
[tex]\[ \text{Bone Density}_2 = \frac{1.7 \, \text{g}}{5.4 \, \text{cm}^3} \approx 0.3148 \, \text{g/cm}^3 \][/tex]

- For Sample 3:
[tex]\[ \text{Bone Density}_3 = \frac{3.6 \, \text{g}}{11.3 \, \text{cm}^3} \approx 0.3186 \, \text{g/cm}^3 \][/tex]

- For Sample 4:
[tex]\[ \text{Bone Density}_4 = \frac{5.2 \, \text{g}}{17.4 \, \text{cm}^3} \approx 0.2989 \, \text{g/cm}^3 \][/tex]

2. Calculate the average bone density:

Now, add the bone densities of all samples and divide by the number of samples:
[tex]\[ \text{Average Bone Density} = \frac{\text{Bone Density}_1 + \text{Bone Density}_2 + \text{Bone Density}_3 + \text{Bone Density}_4}{4} \][/tex]
Substituting the values obtained:
[tex]\[ \text{Average Bone Density} = \frac{0.3009 + 0.3148 + 0.3186 + 0.2989}{4} \approx \frac{1.2332}{4} \approx 0.3083 \, \text{g/cm}^3 \][/tex]

So, the average bone density for the data samples provided is:

[tex]\[ \boxed{0.31 \, \text{g/cm}^3} \][/tex]

This is closest to the given option [tex]\(0.31 \, \text{g/cm}^3\)[/tex].