Sure, let's start by analyzing the given information and then complete the table step-by-step.
From the problem, we know that the amount earned by each worker, [tex]\( A \)[/tex], varies inversely with the number of workers, [tex]\( n \)[/tex]. This relationship is expressed as:
[tex]\[ A = \frac{k}{n} \][/tex]
where [tex]\( k \)[/tex] is a constant. Given that the total labor budget is \[tex]$25,000, this constant \( k \) can be determined by the equation when \( n = 1 \):
\[ k = 25,000 \]
This constant implies that for any number of workers \( n \), the total amount earned by each worker \( A \) times the number of workers \( n \) would always equal \$[/tex]25,000.
Now let's complete the table using the inverse relationship [tex]\( A = \frac{25,000}{n} \)[/tex].
\begin{tabular}{c|c}
[tex]$n$[/tex] & A \\
\hline
1 & 25,000 \\
\hline
2 & 12,500 \\
\hline
3 & \frac{25,000}{3} \approx 8333.33 \\
\hline
5 & \frac{25,000}{5} = 5,000 \\
\hline
10 & \frac{25,000}{10} = 2,500 \\
\hline
25 & \frac{25,000}{25} = 1,000 \\
\hline
50 & \frac{25,000}{50} = 500 \\
\hline
\end{tabular}
Here is the completed table:
\begin{tabular}{c|c}
[tex]$n$[/tex] & A \\
\hline
1 & 25,000 \\
\hline
2 & 12,500 \\
\hline
3 & 8,333.33 \\
\hline
5 & 5,000 \\
\hline
10 & 2,500 \\
\hline
25 & 1,000 \\
\hline
50 & 500 \\
\hline
\end{tabular}
Now, for the second part of the question: we notice that the product [tex]\( A \cdot n \)[/tex] should always equal the total budget, which is \[tex]$25,000. Let's verify this for each entry in the table:
- For \( n = 1 \), \( A = 25,000 \), so \( A \cdot n = 25,000 \cdot 1 = 25,000 \)
- For \( n = 2 \), \( A = 12,500 \), so \( A \cdot n = 12,500 \cdot 2 = 25,000 \)
- For \( n = 3 \), \( A = 8,333.33 \), so \( A \cdot n = 8,333.33 \cdot 3 \approx 25,000 \)
- For \( n = 5 \), \( A = 5,000 \), so \( A \cdot n = 5,000 \cdot 5 = 25,000 \)
- For \( n = 10 \), \( A = 2,500 \), so \( A \cdot n = 2,500 \cdot 10 = 25,000 \)
- For \( n = 25 \), \( A = 1,000 \), so \( A \cdot n = 1,000 \cdot 25 = 25,000 \)
- For \( n = 50 \), \( A = 500 \), so \( A \cdot n = 500 \cdot 50 = 25,000 \)
Hence, we observe that the product \( A \cdot n \) is indeed constant and equals \$[/tex]25,000 for all values of [tex]\( n \)[/tex].
