To determine the weighted average length of the nails from the carpenter's box using the provided data, follow these steps:
1. Identify the Relevant Data:
- For short nails:
- Abundance: [tex]\(70.5\%\)[/tex]
- Length: [tex]\(2.5 \, \text{cm}\)[/tex]
- For medium nails:
- Abundance: [tex]\(19.0\%\)[/tex]
- Length: [tex]\(5.0 \, \text{cm}\)[/tex]
- For long nails:
- Abundance: [tex]\(10.5\%\)[/tex]
- Length: [tex]\(7.5 \, \text{cm}\)[/tex]
2. Convert Percentages to Decimal Form:
- Short nail abundance: [tex]\(70.5\% = 0.705\)[/tex]
- Medium nail abundance: [tex]\(19.0\% = 0.190\)[/tex]
- Long nail abundance: [tex]\(10.5\% = 0.105\)[/tex]
3. Calculate the Total Abundance:
[tex]\[
\text{Total Abundance} = 0.705 + 0.190 + 0.105 = 1.00
\][/tex]
4. Compute the Sum of Weighted Lengths:
- The weighted length for short nails:
[tex]\[
(0.705 \times 2.5) = 1.7625 \, \text{cm}
\][/tex]
- The weighted length for medium nails:
[tex]\[
(0.190 \times 5.0) = 0.95 \, \text{cm}
\][/tex]
- The weighted length for long nails:
[tex]\[
(0.105 \times 7.5) = 0.7875 \, \text{cm}
\][/tex]
5. Sum the Weighted Lengths:
[tex]\[
1.7625 + 0.95 + 0.7875 = 3.5 \, \text{cm}
\][/tex]
6. Calculate the Weighted Average Length:
Since the total abundance is 1.00, the weighted average length is the sum of the weighted lengths:
[tex]\[
\text{Weighted Average Length} = \frac{3.5}{1.00} = 3.5 \, \text{cm}
\][/tex]
Thus, the weighted average length of a nail from the carpenter's box is [tex]\( 3.5 \, \text{cm} \)[/tex].
