Suppose that the number of cars manufactured at an automobile plant varies jointly as the number of workers and the time they work. If 340 workers can produce 204 cars in 4 hours, find the number of workers that can produce 180 cars in 3 hours.
c = kwt
so,
c/(wt) = k, a constant
You want w such that
180/(3w) = 204/(340*4)
60/w = 3/20
w = 400
Suppose that the number of cars manufactured at an automobile plant varies jointly as the number of workers and the time they work. If 340 workers can produce 204 cars in 4 hours, find the number of workers that can produce 180 cars in 3 hours.
Well, it looks like we're dealing with a joint variation problem here, which means we have to consider both the number of workers and the time they work.
Let's start by finding the constant of variation, which we'll call k. Since 340 workers can produce 204 cars in 4 hours, we can set up the equation:
340w(t) = k
where w represents the number of workers and t represents the time in hours.
Now, let's use this information to find the value of k. We know that with 340 workers and 4 hours, they produce 204 cars, so we can write the equation:
340 * 4 = k * 204
Simplifying this equation, we find:
1360 = 204k
Dividing both sides by 204, we can determine that k = 6.67 (approximately).
Now that we have the constant of variation, we can use it to solve the second part of the problem. We're asked to find the number of workers that can produce 180 cars in 3 hours. Let's call this unknown quantity x:
x * 3 = 6.67 * 180
120x = 1200
Dividing both sides by 120, we find that x = 10
Therefore, 10 workers can produce 180 cars in 3 hours.
To solve this problem, we first need to understand the concept of joint variation. When two variables vary directly, it means that as one variable increases, the other variable also increases, and vice versa. In this case, the number of cars manufactured at the automobile plant varies jointly as the number of workers and the time they work.
Let's assume that the number of cars manufactured (C) is directly proportional to the number of workers (W) and the time they work (T). Mathematically, we can express this relationship as:
C = k * W * T
Where k is the constant of variation.
Given that 340 workers can produce 204 cars in 4 hours, we can substitute these values into the equation to find the value of k:
204 = k * 340 * 4
Now, let's solve for k:
k = 204 / (340 * 4)
k = 0.15
Now that we have the value of k, we can use it to solve the second part of the question. We need to find the number of workers that can produce 180 cars in 3 hours.
Using the same equation, we have:
180 = 0.15 * W * 3
Now, let's solve for W:
W = 180 / (0.15 * 3)
W = 400
Therefore, 400 workers can produce 180 cars in 3 hours.