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 long will it take 160 workers to build 16 miles of highway?
t = k L/n
6 = k 4/100
so
k = 150
t = 150 (16/160) = 15 weeks
Well, let's do some clown math here! If it takes 100 workers 6 weeks to build 4 miles of highway, we can determine the proportionality constant.
First, we'll say that the time it takes to build the highway (T) is directly proportional to the length (L) and inversely proportional to the number of workers (W).
So, T ∝ (L/W)
From the given information, we can plug in the values:
6 weeks ∝ (4 miles / 100 workers)
To find the constant of proportionality (k), we can cross multiply:
6 weeks * 100 workers = 4 miles * k
600 weeks * workers = 4 miles * k
Now that we found the constant k, we can use it to determine how long it'll take 160 workers to build 16 miles of highway:
T ∝ (L/W)
T = (L/W) * k
T = (16 miles / 160 workers) * 600 weeks * workers / 4 miles
T = (16 miles / 160 workers) * 150 weeks
T = 1 week
Did you expect that? Well, 160 workers are twice as many as 100 workers, and they have twice as many miles to build, so the time it takes will be the same! It'll still take 1 week to build 16 miles of highway. Not bad, right?
To solve this problem, we can set up a proportion using the given information.
Let's denote:
- t1 as the unknown time it takes for 160 workers to build 16 miles of highway,
- w1 as the unknown number of workers,
- d1 as the unknown distance (16 miles),
- t2 as the given time (6 weeks),
- w2 as the given number of workers (100 workers), and
- d2 as the given distance (4 miles).
Using the given information, we can set up the proportion:
t1 / t2 = (d1 / d2) * (w2 / w1)
Substituting the given values, we have:
t1 / 6 = (16 / 4) * (100 / 160)
Simplifying the equation, we get:
t1 / 6 = 4 * 0.625
t1 / 6 = 2.5
To solve for t1, we can multiply both sides of the equation by 6:
t1 = 2.5 * 6
t1 = 15
Therefore, it will take 160 workers approximately 15 weeks to build 16 miles of highway.
To solve this problem, we can use the concept of direct and inverse variation.
Let's first determine the constant of variation, which represents the relationship between time, length of the highway, and the number of workers.
We know that it takes 100 workers 6 weeks to build 4 miles of highway. Using this information, we can set up a proportion to find the constant of variation:
(time1)/(time2) = (length1)/(length2) * (workers2)/(workers1)
Let's plug in the given values:
6 weeks / time2 = 4 miles / 16 miles * 100 workers / 160 workers.
Simplifying the equation, we have:
6 weeks / time2 = 1/4 * 100/160.
To solve for time2, we can cross-multiply:
6 weeks * 4 * 160 = time2 * 1 * 100.
Simplifying further, we get:
3840 = 100 * time2.
Dividing both sides by 100:
time2 = 3840 / 100.
time2 = 38.4 weeks.
Therefore, it will take 160 workers approximately 38.4 weeks to build 16 miles of highway.