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Proportional relationships are fundamental in understanding how different quantities relate to each other! In such relationships, one quantity is directly proportional to another.
If we say "A varies directly with B," it means as B goes up, A goes up in a consistent way, and similarly, if B goes down, A will go down with it. This kind of association is typically captured by the equation:

• \( A = kB \)

Here, \( k \) is a special number called the 'constant of proportionality,' which determines how much A will change when B changes.
To illustrate, think of A as pollution produced and B as the population count. As a city's population increases, the amount of pollution it generates rises, assuming of course that the factors affecting pollution remain the same.

The constant of proportionality, represented by the symbol \( k \), feeds into the magic of direct variation equations. This constant tells you exactly how one variable is changing in relation to another.
It's like the conversion rate in a currency exchange, taking you from one currency to another consistently.

Finding the Constant

To find the constant of proportionality, you need a known pair of values. For Kansas City, the population was \( 442,000 \) and generated \( 260,000 \) tons of pollutants.

• Substitute these into the equation \( P = kN \):
• \( 260,000 = k \cdot 442,000 \)

By solving \( k \), you get the specific rate at which pollution increases per person in that population:

• \[ k = \frac{260,000}{442,000} \approx 0.588 \]

This means for every person in the city, about \( 0.588 \) tons of pollution is associated.

The relationship between population and pollution is a common real-world example of direct variation. As populations of cities and regions grow, so does the amount of pollution, assuming other variables stay consistent.
This is why understanding direct variation in this context is important for environmental planning and management.
Take Kansas City and St. Louis for instance, where population numbers were used to estimate pollution levels.

Applying the Relationship

With a calculated constant of proportionality, you can easily project potential pollution levels for other populations.
This helps in environmental assessments, allowing cities to plan better for pollution control and infrastructure:

• For St. Louis with a population of \( 348,000 \), and using \( k \approx 0.588 \), the projected pollution is:
• \( P' = 0.588 \times 348,000 = 204,624 \)

Rounding gives an easy-to-interpret figure, making these projections handy for decision-makers and environmentalists.