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].
