The amount P of pollution varies directly with the population N of people. City A has a population of 468,000 and produces 260,000 tons of pollutants. Find how many tons of pollution we should expect City B to produce, if we know that it's population is 362,000.
To solve this problem, we can use the concept of direct variation. Direct variation occurs when two quantities are related in such a way that when one quantity changes, the other quantity changes proportionally.
In this case, we are given that the amount of pollution (P) varies directly with the population (N) of people. Mathematically, we can represent this relationship as:
P = kN
where k is the constant of variation.
We are given the population (N) and amount of pollution (P) for City A, which allows us to find the value of k. Let's substitute the values into the equation:
260,000 = k * 468,000
To solve for k, we divide both sides of the equation by 468,000:
k = 260,000 / 468,000
Now that we have the value of k, we can use it to find the expected tons of pollution for City B. We know that City B has a population of 362,000, so we can substitute this value into our equation:
P = (260,000 / 468,000) * 362,000
Now we can calculate the expected tons of pollution for City B.
To find out how many tons of pollution we should expect City B to produce, we can set up a proportion based on the direct variation relationship between population and pollution.
The direct variation equation is P = kN, where P is the amount of pollution, N is the population, and k is the constant of variation.
First, we need to find the value of k using the given information for City A.
P₁ = 260,000 tons (pollution in City A)
N₁ = 468,000 people (population in City A)
Now, we can set up the proportion:
P₁/N₁ = P₂/N₂
Substituting the known values:
260,000/468,000 = P₂/362,000
To solve for P₂, we can cross-multiply:
260,000 * 362,000 = 468,000 * P₂
94,120,000,000 = 468,000 * P₂
Now, divide both sides by 468,000 to isolate P₂:
94,120,000,000/468,000 = P₂
This gives us:
P₂ ≈ 201,940 tons
Therefore, we can expect City B to produce approximately 201,940 tons of pollution.
Use a proportion. Cross multiply and solve for x.
468,000/260,000 = 362,000/x