Let t represent the amount of time required
Let l represent the length of the highway
Let n represent the number of workers
Let k represent the constant of proportionality.
If the amount of time it takes to build a highway varies directly with the length of the highway, then the relationship would be
t = kl
If the amount of time it takes to build a highway varies inversely with the number of workers, then the relationship is
t = k/n
If we combine both equations, the we would have
t = kl/n
If it takes 150 workers 14 weeks to build 12 miles of Highway, then
n = 150, t = 14 and l = 12
Substituting these values into the equation, we have
14 = k * 12/150
14 = 0.08k
k = 14/0.08
k = 175
Thus, the equation is
t = 175l/n
Thus, if l = 15, t = 21, then the number of workers needed, n would be
21 = 175 * 15/n
21n = 2625
n = 2625/21
n = 125
Thus, 125 workers would be needed to build 15 miles of Highway in 21 weeks