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 150 workers 14 weeks to build 12 miles of highway. How many workers would be needed to build 15 miles of highway in 21 weeks?
To solve this problem, we can set up a proportion based on the given information.
Let x be the number of workers needed to build 15 miles of highway in 21 weeks.
Using the direct and inverse variation relationships:
(Workers) = k × (Miles) / (Weeks)
Where k is a constant of variation that we need to find.
From the given data, we know that it takes 150 workers 14 weeks to build 12 miles of highway. Therefore, we can set up the following proportion:
150 = k × 12 / 14
Solving for k:
150 = 12k / 14
k = (150 * 14) / 12
k = 175
Now that we have found the constant of variation, we can set up the proportion for the new situation:
x = 175 * 15 / 21
Solving for x:
x = 175 * 15 / 21
x = 125
Therefore, 125 workers would be needed to build 15 miles of highway in 21 weeks.