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].
