Answer:
The amount of time it would take 160 workers to build 16 miles of Highway is;
[tex]15\text{ weeks}[/tex]Explanation:
Given that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers.
Let t represent the time in weeks, l represents the length of the highway in miles, and n represent the number of workers;
So, we have;
[tex]\begin{gathered} t\propto\frac{l}{n} \\ t=k\frac{l}{n} \\ k=\frac{nt}{l} \end{gathered}[/tex]Given;
it takes 100 workers 6 weeks to build 4 miles of Highway
[tex]\begin{gathered} n=100 \\ t=6 \\ l=4 \\ k=\frac{nt}{l} \\ k=\frac{100\times6}{4} \\ k=150 \end{gathered}[/tex]so, for 160 workers to build 16 miles of Highway the time it will take is;
[tex]\begin{gathered} n=160 \\ l=16 \\ k=150 \\ t=k\frac{l}{n} \\ t=150\times\frac{16}{160} \\ t=\frac{150}{10} \\ t=15 \end{gathered}[/tex]Therefore, the amount of time it would take 160 workers to build 16 miles of Highway is;
[tex]15\text{ weeks}[/tex]