To find the weighted average mass of a penny in the sample, we need to combine the contributions of the two different types of pennies — those made before 1982 and those made after 1982. The weighted average takes into account the mass of each type of penny and their respective abundances in the sample.
Here are the steps to calculate the weighted average mass:
1. Convert Abundances to Proportions
- Pre-1982 abundance is given as 36%. In decimal form, this is [tex]\(36 / 100 = 0.36\)[/tex].
- Post-1982 abundance is given as 64%. In decimal form, this is [tex]\(64 / 100 = 0.64\)[/tex].
2. Identify the Mass of Each Type
- The mass of a pre-1982 penny is 3.1 grams.
- The mass of a post-1982 penny is 2.5 grams.
3. Apply the Weighted Average Formula
The weighted average mass [tex]\( \text{WAM} \)[/tex] is calculated using the formula:
[tex]\[
\text{WAM} = (\text{Proportion of Pre-1982} \times \text{Mass of Pre-1982}) + (\text{Proportion of Post-1982} \times \text{Mass of Post-1982})
\][/tex]
4. Substitute the Values
[tex]\[
\text{WAM} = (0.36 \times 3.1) + (0.64 \times 2.5)
\][/tex]
5. Calculate Each Term
- For the pre-1982 pennies: [tex]\(0.36 \times 3.1 = 1.116\)[/tex]
- For the post-1982 pennies: [tex]\(0.64 \times 2.5 = 1.6\)[/tex]
6. Sum the Contributions
[tex]\[
\text{WAM} = 1.116 + 1.6 = 2.716
\][/tex]
So, the weighted average mass of a penny in the sample, taking into account the different abundances and masses of the pre-1982 and post-1982 pennies, is 2.716 grams.
