Suppose 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. Suppose also that it takes 200 workers 3 weeks to build 2 miles of highway. How many workers would be needed to build 5 miles of highway in 6 weeks?
Let h be the length of the highway (in miles), w be the number of workers, and t be the time it takes to build the highway (in weeks). Since 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, we can write:
t = k * (h / w),
where k is the constant of variation. We can now use the given information to find k. We know that for h = 2 and w = 200, it takes t = 3 weeks to build the highway:
3 = k * (2 / 200).
k = 3 / (2 / 200),
k = 300.
Now we can use the same formula to find the number of workers needed to build 5 miles of highway in 6 weeks:
6 = 300 * (5 / w).
w = 300 * (5 / 6),
w = 250.
So, 250 workers would be needed to build 5 miles of highway in 6 weeks.