Problem 40 - 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.

Key Concepts

Constant of Variation

In direct variation, the constant of variation is a key element. It acts as a consistent multiplier that ties two quantities together. Imagine it as the glue that maintains a fixed ratio between pollution and population in this context. For example, when we say pollutants vary directly with the population, we mean there is a specific, unchanging number called the constant of variation. This number tells us how much pollution is produced per person in the population.

The equation for direct variation typically looks like this:

• Pollution ( \(P\) ) = Constant of Variation ( \(k\) ) \( \times \) Population ( \(P_{op}\) )

Finding the constant of variation involves substituting values into this equation. Given a specific population and the associated pollution level, we compute \(k\) by dividing the pollution by the population. In our example, a city of 500,000 produces 800,000 tons of pollutants, resulting in a \(k\) of 1.6. This fundamental calculation enables us to predict pollution levels for any population size using the constant of variation.

Proportional Relationships

Proportional relationships exist when two quantities maintain a constant ratio. These relationships reflect a unique type of symmetry in mathematics, ensuring that the corresponding division results in the same number every time. Direct variation is a particular instance of proportional relationships where one variable increases or decreases in direct proportion to another.

In real life, recognizing proportional relationships helps in predicting outcomes. For instance, in our pollution problem, knowing the constant allows us to surmise that if the population doubles, so will the pollution. These predictable patterns help us understand and manage resource allocation effectively.

Using Proportional Relationships

To apply proportional relationships:

• Identify the constant multiplier between the two quantities.
• Ensure that changes in one result in proportional changes in the other.

A city with a population that increases from 500,000 to 1,000,000 expects its pollution to increase accordingly, as signified by the unchanged constant multiplier, 1.6, in this case.

Pollution Calculation

Pollution calculation through direct variation offers a simplified approach to understanding environmental impacts. In our example, knowing the direct relationship between population size and pollution enables cities to estimate contamination levels quickly. The calculation starts from the known constant of variation, discovered earlier.

For a city of 1,000,000 people, we apply the equation:

• Pollution ( \(P\) ) = 1.6 \( \times \) 1,000,000

Demonstrating a pollution output of 1,600,000 tons. This straightforward method is not only beneficial for city planners but also for environmental advocates and policy-makers, striving to mitigate pollution.

Practical Implications

Understanding how to calculate based on population assists in:

• Predicting future pollution levels based on projected population growth.
• Implementing effective regulatory measures and environmental policies.

Ultimately, pollution calculations using direct variation aid in fostering sustainable urban development and environmental stewardship.