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 100 workers 6 weeks to build 4 miles of highway. How many workers would be needed to build 16 miles of highway in 15 weeks?
t = k d / n
6 = k 4 / 100
k = 600/4 = 150
so
t = 150 * d / n
then
15 = 150 * 16 / n
1/10 = 16/n
= 16/160
To solve this problem, we can use the proportionality equation:
time = k * (length / workers)
where k is the constant of proportionality.
Given that it takes 100 workers 6 weeks to build 4 miles of highway, we can calculate k:
6 = k * (4 / 100)
K = 150
Now, we can use k to find the number of workers needed to build 16 miles of highway in 15 weeks:
time = 150 * (16 / workers)
Substituting the given values, we have:
15 = 150 * (16 / workers)
To solve for workers, we'll isolate the variable:
15 * workers = 150 * 16
15 * workers = 2400
Dividing both sides by 15, we get:
workers = 2400 / 15
workers = 160
Therefore, 160 workers would be needed to build 16 miles of highway in 15 weeks.
To solve this problem, we can use the concept of direct and inverse variation. Direct variation means that two variables increase or decrease at the same rate, while inverse variation means that as one variable increases, the other variable decreases.
Let's break down the given information:
1. The amount of time it takes to build a highway varies directly with the length of the highway: This means that as the length of the highway increases, the time it takes to build it also increases proportionally.
2. The amount of time it takes to build a highway varies inversely with the number of workers: This means that as the number of workers decreases, the time it takes to build the highway increases proportionally.
Given: 100 workers, 6 weeks, and 4 miles of highway.
To find the number of workers needed to build 16 miles of highway in 15 weeks, we can set up a proportion using the direct and inverse variation relationships:
Number of Workers / Time = Length / Constant
First, let's find the constant value using the given information:
100 workers * 6 weeks = 4 miles * constant
600 = 4 * constant
constant = 600 / 4
constant = 150
Now, we can set up the proportion using the constant value:
Number of Workers / 15 weeks = 16 miles / 150
To solve for the number of workers, cross-multiply and then divide:
Number of Workers = (15 weeks * 16 miles) / 150
Number of Workers = 240 / 150
Number of Workers = 1.6
Since the number of workers cannot be a decimal, we need to round it up to the nearest whole number since you can't have a fraction of a worker.
Therefore, you would need 2 workers to build 16 miles of highway in 15 weeks.