Problem 79 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond's oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content \(t\) days after organic waste has been dumped into the pond is given by $$ P=100\left(\frac{t^{2}+10 t+100}{t^{2}+20 t+100}\right) $$ percent of its normal level. Find \(t\) corresponding to an oxygen content of \(80 \%\) and interpret your results.

Step-by-Step Solution

Answer

The feasible solution is \(t=0\), meaning that the oxygen content in the pond is at \(80\%\) of its normal level immediately after the organic waste has been dumped. This indicates that the pond's oxygen content is affected quickly, but will eventually return to its original level over time.

Step 1: Set up the equation

We have the oxygen content function: \[P(t) = 100\left(\frac{t^{2}+10 t+100}{t^{2}+20 t+100}\right)\] Now, we want to find \(t\) when \(P(t) = 80\). So, set \(P(t)\) equal to \(80\): \[80 = 100\left(\frac{t^{2}+10 t+100}{t^{2}+20 t+100}\right)\]

Step 2: Solve for \(t\)

To solve for \(t\), let's first simplify the equation by dividing both sides by 100: \[0.8 = \frac{t^{2}+10 t+100}{t^{2}+20 t+100}\] Now, to get rid of the fraction, we can multiply both sides by the denominator \(t^{2} + 20t + 100\): \[0.8(t^{2} + 20t + 100) = t^{2} + 10t + 100\] Expanding and simplifying the equation, we get: \[0.2t^{2} + 6t = 0\] Now, we can factor out \(t\): \[t(0.2t + 6) = 0\] This gives us two potential solutions for \(t\): \(t = 0\) and \(t = -30\).

Step 3: Find the feasible solution and interpret the result

Since \(t\) represents the number of days after organic waste has been dumped into the pond, it must be positive. Therefore, we can discard the solution \(t = -30\) as it doesn't make sense in this context. Thus, the feasible solution is \(t=0\). This means that the oxygen content in the pond is at \(80\%\) of its normal level immediately after the organic waste has been dumped. This implies that the pond's oxygen content is affected quite quickly, but over time, it will eventually return to its original level.

Key Concepts

Oxygen Content

When we talk about oxygen content in a pond, we refer to the amount of dissolved oxygen as a percentage of its natural level. This is vital for aquatic life, as organisms such as fish and plants rely on it to survive.
The oxygen content can often change due to environmental factors or human activities. In the scenario we're looking at, organic waste is dumped into a pond, and its decomposition begins to quickly lower the available oxygen. To mathematically represent this change, a formula can be used:

• The formula shows how the oxygen percentage varies over time.
• The variable "t" represents the number of days after waste introduction.
• The function given calculates this percentage based on the changes over time within the pond's ecosystem.

Understanding how oxygen levels change helps us predict and potentially mitigate negative effects on aquatic ecosystems.

Oxidation Process

The oxidation process is a chemical reaction that involves the loss of electrons, often resulting in a compound becoming oxidized. In environmental terms, specifically within a water body, this becomes crucial once organic waste is added. The organic material serves as food for bacteria, which consume oxygen as they decompose the matter.
As this biological oxidation proceeds:

• Oxygen levels drop because microorganisms use it for decomposition.
• If these levels drop too low, it can create a harmful environment for aquatic creatures.
• Fortunately, ecosystems have natural processes to slowly reinstate oxygen levels over time.

This complex interplay underscores how essential it is for us to monitor such changes. By understanding oxidation in a pond, we can take measures to prevent adverse effects on aquatic life.

Environmental Mathematics

Environmental mathematics combines math techniques with environmental science to assess and solve ecological problems. In our example, using a formula to track oxygen content over time helps manage the effects of pollution in natural water bodies. By solving equations that show how variables transform within ecosystems:

• Mathematics provides quantitative insights into environmental changes.
• This understanding can assist researchers and policymakers in crafting solutions for ecosystem management.
• Analyzing functions like our oxygen content equation is essential for predicting future states and conditions.

Through applied mathematics, we convert abstract issues into data that can guide effective environmental strategies.