To determine the type of metal for the unknown sample, we need to follow these steps:
1. Calculate the density of the unknown metal sample.
2. Determine the densities of the known metal samples.
3. Compare the density of the unknown sample with the densities of the known samples to find the closest match.
### Step 1: Calculate the density of the unknown metal sample
The density of a material can be calculated using the formula:
[tex]\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \][/tex]
For the unknown metal sample:
[tex]\[ \text{Mass} = 33.5 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 3.2 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{33.5 \, \text{g}}{3.2 \, \text{cm}^3} = 10.46875 \, \text{g/cm}^3 \][/tex]
### Step 2: Calculate the densities of the known metal samples
Using the same density formula for each known metal:
- Aluminum:
[tex]\[ \text{Mass} = 14.6 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 5.4 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{14.6 \, \text{g}}{5.4 \, \text{cm}^3} \approx 2.70 \, \text{g/cm}^3 \][/tex]
- Iron:
[tex]\[ \text{Mass} = 33.1 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 4.2 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{33.1 \, \text{g}}{4.2 \, \text{cm}^3} = 7.88 \, \text{g/cm}^3 \][/tex]
- Lead:
[tex]\[ \text{Mass} = 35.2 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 3.1 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{35.2 \, \text{g}}{3.1 \, \text{cm}^3} \approx 11.35 \, \text{g/cm}^3 \][/tex]
- Magnesium:
[tex]\[ \text{Mass} = 10.6 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 6.1 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{10.6 \, \text{g}}{6.1 \, \text{cm}^3} \approx 1.74 \, \text{g/cm}^3 \][/tex]
- Silver:
[tex]\[ \text{Mass} = 47.2 \, \text{g} \][/tex]
[tex]\[ \text{Volume} = 4.5 \, \text{cm}^3 \][/tex]
[tex]\[ \text{Density} = \frac{47.2 \, \text{g}}{4.5 \, \text{cm}^3} \approx 10.49 \, \text{g/cm}^3 \][/tex]
### Step 3: Compare the densities
We now compare the density of the unknown sample (10.46875 g/cm³) with the densities of the known samples:
- Aluminum: 2.70 g/cm³
- Iron: 7.88 g/cm³
- Lead: 11.35 g/cm³
- Magnesium: 1.74 g/cm³
- Silver: 10.49 g/cm³
The density of the unknown sample (10.46875 g/cm³) is closest to the density of silver (10.49 g/cm³).
Therefore, the type of metal for the unknown sample is most likely silver. Among the given choices, the correct answer is not listed directly; however, the density closest in the provided table to that of the unknown sample would be that of silver.
