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Algebraic functions are mathematical expressions that involve variables and constants. Here, we deal with an algebraic function that models the cost to remove a percentage of waste from a river. The formula given is \( C(x) = \frac{80x}{100 - x} \). This function shows how the cost \( C \) varies with the percentage of waste removed \( x \). Note how the denominator \( (100 - x) \) reflects an increasing difficulty in removing waste as \( x \) approaches 100%. This type of function is useful because it allows us to predict costs based on different percentages of waste removal.

Cost analysis involves evaluating all factors that contribute to the total cost of a project. In this exercise, the formula \( C(x) = \frac{80x}{100 - x} \) lets us calculate costs for specific percentages of waste removal.
For instance, removing 20% of the waste costs \( 20,000 \) dollars, calculated by substituting \( x = 20 \) into the formula:
\[ C(20) = \frac{80 \times 20}{100 - 20} = \frac{1600}{80} = 20 \text{(in units of } \$1000) = 20,000 \$ \text{(rounded to nearest thousand)} \ \] Similarly, other percentages can be computed by substituting the corresponding \( x \) values into the function.
This analysis helps in budgeting and understanding financial requirements for different levels of environmental cleanup.

Environmental mathematics combines mathematical concepts with environmental issues to find solutions. The cost model for pollution removal, \( C(x) = \frac{80x}{100 - x} \), is an example of how math can assist in tackling pollution. By understanding this function, city planners can make informed decisions about managing pollution within budget constraints.
Sustainable practices often involve balancing costs with environmental benefits. Mathematics provides the tools needed to optimize such efforts, ensuring efficient and effective use of resources to achieve desired levels of waste reduction.

Budget constraints limit the amount of money available for projects. In this exercise, the city has a budget of \$320,000 for river cleanup. To determine what percentage of waste can be removed under this budget, solve for \( x \) in the equation \( \frac{80x}{100 - x} = 320 \).
Simplifying, we find:
\[ \frac{80x}{100 - x} = 320 \] First, multiply both sides by \( 100 - x \) to clear the denominator:
\[ 320(100 - x) = 80x \] Then simplify and solve for \( x \):
\[ 32000 - 320x = 80x \] Combine like terms:
\[ 32000 = 400x \] Finally, solve for \( x \):
\[ x = 80 \] This means with the given budget, the city can afford to remove 80% of the waste.
Understanding budget constraints ensures that projects remain financially viable while achieving environmental goals.