Problem 84 - Algebra and Trigonometry

Question

Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that \(10 \%\) of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after \(n\) years is given by the recursively defined sequence \(p_{0}=250 \quad p_{n}=0.9 p_{n-1}+15\) Determine the amount of pollutant in the lake after 2 years. That is, determine \(p_{2}\).

Step-by-Step Solution

Answer

231 tons

Step 1: Understand the initial conditions

The initial amount of pollutant in the lake is given by \( p_{0} = 250 \). This is the starting point from which the calculations will be made.

Step 2: Identify the recurrence relation

The amount of pollutant in the lake each year is determined by the recurrence relation: \( p_{n} = 0.9 p_{n-1} + 15 \). This means that each year, 90% of the previous year's pollutant remains, and an additional 15 tons of new pollutant is added.

Step 3: Calculate the pollutant after one year

Using the recurrence relation and the initial condition, calculate the amount of pollutant after one year (\( p_{1} \)): \[ p_{1} = 0.9 p_{0} + 15 \] \[ p_{1} = 0.9 (250) + 15 \] \[ p_{1} = 225 + 15 \] \[ p_{1} = 240 \] So, the pollutant after one year is 240 tons.

Step 4: Calculate the pollutant after two years

Now, use the recurrence relation to calculate the amount of pollutant after two years (\( p_{2} \)): \[ p_{2} = 0.9 p_{1} + 15 \] \[ p_{2} = 0.9 (240) + 15 \] \[ p_{2} = 216 + 15 \] \[ p_{2} = 231 \] Therefore, the amount of pollutant in the lake after two years is 231 tons.

Key Concepts

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Initial conditions are the starting values given in a recurrence relation problem. These values help us begin the sequence calculations. In our exercise, the initial condition is given as the amount of pollutant in the lake, which is 250 tons. This is represented mathematically as:

• \(p_{0} = 250\)

This value is crucial because it serves as the base from which all future calculations start. Without it, we wouldn't be able to determine the levels of pollutants in subsequent years. Initial conditions are like the foundation of a building; they provide the starting point for constructing the sequence.

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A recurrence relation is a formula that defines each term of a sequence using the preceding terms. In simpler terms, it's a way to calculate the next value based on the current value. In our exercise, the recurrence relation is defined as:

• \(p_{n} = 0.9 p_{n-1} + 15\)

This means that at any given year, 90% of the previous year's pollutant remains, and an additional 15 tons of new pollutants are introduced. Recurrence relations are essential in modeling real-world phenomena where future states depend on past states, making them widely used in areas like population studies, finance, and environmental science.

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Sequence calculation involves using the initial conditions and the recurrence relation to find the values of the sequence step-by-step. In our problem, we start with the initial condition \(p_{0} = 250\). We then use the recurrence relation to find the amount of pollutant in subsequent years.

For the first year:

• \(p_{1} = 0.9 p_{0} + 15\)
• \(p_{1} = 0.9 \times 250 + 15 = 240\)

For the second year:

• \(p_{2} = 0.9 p_{1} + 15\)
• \(p_{2} = 0.9 \times 240 + 15 = 231\)

Thus, the pollutant levels for the first and second years are 240 tons and 231 tons, respectively. Sequence calculation allows us to predict future states based on an established pattern.

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Environmental pollution refers to the introduction of harmful substances or products into the environment, which negatively impact human health and ecosystems. In this exercise, the pollutants come from industrial waste. Pollution control laws play a critical role in managing the amount of pollutants entering places like Maple Lake.

However, despite controls, factors like solar oxidation, which neutralizes 10% of the pollutants, and the introduction of an additional 15 tons of pollutants annually still affect the lake.

Understanding concepts like recurrence relations and sequence calculations helps scientists and policymakers predict the outcomes of various environmental scenarios, leading to more effective pollution control strategies.