To solve this problem, we must group the class periods according to their density values relative to the true density of aluminum, which is [tex]$2.7 \, \text{g/ml}$[/tex]. Here’s a detailed step-by-step solution:
1. List the given density values for each class period:
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(2.35 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
2. Identify the true density value:
- True density of aluminum: [tex]\(2.7 \, \text{g/ml}\)[/tex]
3. Compare each class period's density value with the true density:
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex] (greater than [tex]\(2.7 \, \text{g/ml}\)[/tex])
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex] (greater than [tex]\(2.7 \, \text{g/ml}\)[/tex])
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex] (less than [tex]\(2.7 \, \text{g/ml}\)[/tex])
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(2.35 \, \text{g/ml}\)[/tex] (less than [tex]\(2.7 \, \text{g/ml}\)[/tex])
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex] (greater than [tex]\(2.7 \, \text{g/ml}\)[/tex])
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex] (greater than [tex]\(2.7 \, \text{g/ml}\)[/tex])
4. Group the class periods based on whether their recorded density is under the true density or over the true density:
- Values under the true density:
- [tex]\(3^{\text{rd}}\)[/tex] hour: [tex]\(1.9 \, \text{g/ml}\)[/tex]
- [tex]\(4^{\text{th}}\)[/tex] hour: [tex]\(2.35 \, \text{g/ml}\)[/tex]
- Values over the true density:
- [tex]\(1^{\text{st}}\)[/tex] hour: [tex]\(3.1 \, \text{g/ml}\)[/tex]
- [tex]\(2^{\text{nd}}\)[/tex] hour: [tex]\(3.05 \, \text{g/ml}\)[/tex]
- [tex]\(5^{\text{th}}\)[/tex] hour: [tex]\(4.2 \, \text{g/ml}\)[/tex]
- [tex]\(6^{\text{th}}\)[/tex] hour: [tex]\(4.0 \, \text{g/ml}\)[/tex]
Given these groupings, the correct answer among the provided options is:
b. Group: [tex]\(3^{\text{rd}}\)[/tex] and [tex]\(4^{\text{th}}\)[/tex] hours, with values under the true density
Group: [tex]\(1^{\text{st}} \cdot 2^{\text{nd}} \cdot 5^{\text{th}}\)[/tex] and [tex]\(6^{\text{th}}\)[/tex] hours, with values over the true density
Therefore, the answer to the question is:
Option (b) is correct.
