Select the correct answer.

A student has samples of five different types of metal. She measures the mass and volume of each sample. The measurements she took are in the table.

\begin{tabular}{|l|c|c|}
\hline
Type of Metal & \begin{tabular}{c}
Mass \\
[tex]$(g)$[/tex]
\end{tabular} & \begin{tabular}{c}
Volume \\
[tex]$\left(cm^3\right)$[/tex]
\end{tabular} \\
\hline
aluminum & 14.6 & 5.4 \\
\hline
iron & 33.1 & 4.2 \\
\hline
lead & 35.2 & 3.1 \\
\hline
magnesium & 10.6 & 6.1 \\
\hline
silver & 47.2 & 4.5 \\
\hline
\end{tabular}

The student also has another piece of metal. She knows it is the same type of metal as one of her other samples, but she doesn't know which one it is. She finds that its mass is 33.5 grams and its volume is 3.2 centimeters [tex]${ }^3$[/tex]. What type of metal is the unknown sample?

A. aluminum 2.70
B. iron 7.87
C. lead 11.34
D. magnesium 1.74



Answer :

To determine the type of metal of the unknown sample, we need to calculate its density and compare it to the densities of the known metals listed.

1. Density Calculation for the Unknown Sample:
- Mass of the unknown sample: [tex]\(33.5 \, \text{g}\)[/tex]
- Volume of the unknown sample: [tex]\(3.2 \, \text{cm}^3\)[/tex]
[tex]\[ \text{Density of the unknown sample} = \frac{\text{mass}}{\text{volume}} = \frac{33.5}{3.2} \approx 10.46875 \, \text{g/cm}^3 \][/tex]

2. Density Calculation for Each Known Metal:
- Aluminum:
[tex]\[ \text{Density of aluminum} = \frac{14.6 \, \text{g}}{5.4 \, \text{cm}^3} \approx 2.70 \, \text{g/cm}^3 \][/tex]
- Iron:
[tex]\[ \text{Density of iron} = \frac{33.1 \, \text{g}}{4.2 \, \text{cm}^3} \approx 7.88 \, \text{g/cm}^3 \][/tex]
- Lead:
[tex]\[ \text{Density of lead} = \frac{35.2 \, \text{g}}{3.1 \, \text{cm}^3} \approx 11.35 \, \text{g/cm}^3 \][/tex]
- Magnesium:
[tex]\[ \text{Density of magnesium} = \frac{10.6 \, \text{g}}{6.1 \, \text{cm}^3} \approx 1.74 \, \text{g/cm}^3 \][/tex]
- Silver:
[tex]\[ \text{Density of silver} = \frac{47.2 \, \text{g}}{4.5 \, \text{cm}^3} \approx 10.49 \, \text{g/cm}^3 \][/tex]

3. Comparison of Densities:
- Aluminum: [tex]\(2.70 \, \text{g/cm}^3\)[/tex]
- Iron: [tex]\(7.88 \, \text{g/cm}^3\)[/tex]
- Lead: [tex]\(11.35 \, \text{g/cm}^3\)[/tex]
- Magnesium: [tex]\(1.74 \, \text{g/cm}^3\)[/tex]
- Silver: [tex]\(10.49 \, \text{g/cm}^3\)[/tex]

4. Analyzing the Densities:
- The calculated density of the unknown sample is approximately [tex]\(10.47 \, \text{g/cm}^3\)[/tex].
- Among the given metals:
- Aluminum: 2.70
- Iron: 7.87
- Lead: 11.34
- Magnesium: 1.74
- The calculated density of the unknown sample (10.46875) is closest to the density of silver (10.488888888888889).

Based on this comparison, the unknown sample is most likely to be silver.

Therefore, the correct answer is not listed among the choices provided, as silver is not an option. However, given the numerical data:
The closest match for the unknown sample, considering provided options and calculations, resembles that of silver, which would be indicated as the answer if it was listed.