Problem 41 - Beginning and Intermediate Algebra

Question

The amount of pollution produced varies directly as the population. If a city of \(500,000\) people produces \(800,000\) tons of pollutants, how many tons of pollutants would be produced by a city of \(1,000,000\) people?

Step-by-Step Solution

Answer

A city with a population of 1,000,000 would produce 1,600,000 tons of pollutants.

Step 1: Write the direct variation equation

The equation for direct variation can be written as: \(P = kP_{op}\), where: - \(P\) is the amount of pollution produced, - \(k\) is the constant of variation, and - \(P_{op}\) is the population.

Step 2: Find the constant of variation

We are given that a city with a population of 500,000 produces 800,000 tons of pollution. Plug these values into the equation and solve for \(k\): \begin{align*} 800,000 &= k \cdot 500,000 \\ k &= \frac{800,000}{500,000} \\ k &= 1.6 \end{align*} So, the constant of variation is \(k = 1.6\).

Step 3: Write the equation with the constant of variation

Now that we know the constant of variation, we can rewrite the direct variation equation as: \(P = 1.6P_{op}\)

Step 4: Find the pollution for a population of 1,000,000

We are asked to find the amount of pollution produced for a city with a population of 1,000,000. Plug this value into our equation and solve for \(P\): \begin{align*} P &= 1.6 \cdot 1,000,000 \\ P &= 1,600,000 \end{align*} Thus, a city with a population of 1,000,000 would produce 1,600,000 tons of pollutants.