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

Bone Density Data

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

Density [tex]$\left( \frac{g}{cm^3} \right)=$[/tex] Mass [tex]$\left( g \right) /$[/tex] Volume [tex]$\left( cm^3 \right)$[/tex]

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



Answer :

To determine the average bone density for the data samples, we need to follow these steps:

1. Calculate the bone density for each sample using the formula:
[tex]\[ \text{Density} \left( \frac{g}{cm^3} \right) = \frac{\text{Mass} (g)}{\text{Volume} (cm^3)} \][/tex]

Let's start with each sample:

- Sample 1:
- Mass: 6.8 g
- Volume: 22.6 cm[tex]\(^3\)[/tex]
- Density: [tex]\(\frac{6.8 \, g}{22.6 \, cm^3} \approx 0.300885 \, g/cm^3\)[/tex]

- Sample 2:
- Mass: 1.7 g
- Volume: 5.4 cm[tex]\(^3\)[/tex]
- Density: [tex]\(\frac{1.7 \, g}{5.4 \, cm^3} \approx 0.314815 \, g/cm^3\)[/tex]

- Sample 3:
- Mass: 3.6 g
- Volume: 11.3 cm[tex]\(^3\)[/tex]
- Density: [tex]\(\frac{3.6 \, g}{11.3 \, cm^3} \approx 0.318584 \, g/cm^3\)[/tex]

- Sample 4:
- Mass: 5.2 g
- Volume: 17.4 cm[tex]\(^3\)[/tex]
- Density: [tex]\(\frac{5.2 \, g}{17.4 \, cm^3} \approx 0.298851 \, g/cm^3\)[/tex]

2. Now, we'll list all the densities calculated:
- Sample 1: [tex]\(0.300885 \, g/cm^3\)[/tex]
- Sample 2: [tex]\(0.314815 \, g/cm^3\)[/tex]
- Sample 3: [tex]\(0.318584 \, g/cm^3\)[/tex]
- Sample 4: [tex]\(0.298851 \, g/cm^3\)[/tex]

3. To find the average bone density, sum up all the densities and divide by the number of samples:
[tex]\[ \text{Average Density} = \frac{0.300885 + 0.314815 + 0.318584 + 0.298851}{4} \][/tex]

4. When we perform this calculation:
[tex]\[ \text{Average Density} \approx \frac{1.233135}{4} \approx 0.308284 \, g/cm^3 \][/tex]

So, the average bone density for the given samples is approximately [tex]\(0.308 \, g/cm^3\)[/tex].