Problem 11 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Algae is accumulating in a lake at a rate of \(e^{-0.05 t} \sin ^{2} \pi t\). The factor \(e^{-0.05 t}\) reflects declining available oxygen and the factor \(\sin ^{2} \pi t\) reflects diurnal oscillation. Is the amount of algae produced infinite? See Exercise Figure 11.5 .11

Step-by-Step Solution

Answer

The production is not infinite due to the exponential decay factor.

Step 1: Analyze the given equation

The function representing the rate of algae accumulation is given by\[e^{-0.05 t} \sin^2(\pi t)\]This function indicates that the rate depends on two factors: - \( e^{-0.05 t} \), representing the declining oxygen, which is an exponential decay as \( t \) increases.- \( \sin^2(\pi t) \), representing the diurnal oscillations, a periodic function that varies over time.

Step 2: Consider the behavior of individual factors

1. **Oxygen factor:** The term \( e^{-0.05 t} \) declines exponentially as \( t \) approaches infinity, suggesting that the accumulation rate diminishes over time.2. **Diurnal oscillation factor:** The term \( \sin^2(\pi t) \) oscillates between 0 and 1, reflecting a continuous fluctuation, but does not itself tend to increase or decrease over time.

Step 3: Assess the long-term product of the two factors

As seen above, \( e^{-0.05 t} \) tends toward 0 as \( t \) goes to infinity, and \( \sin^2(\pi t) \) oscillates between 0 and 1. The overall product \( e^{-0.05 t} \sin^2(\pi t) \) will thus tend to zero because the exponential decay dominates the oscillating part in the long run.

Step 4: Conclusion about the total algae production

Even though the algae production function is periodic and fluctuating due to \( \sin^2(\pi t) \), the presence of the exponentially decaying factor \( e^{-0.05 t} \) ensures that the algae production rate diminishes over time. This means no infinite accumulation because the function approaches zero.

Key Concepts

Exponential Decay

In mathematics, exponential decay describes the process of a quantity decreasing at a rate proportional to its current value. This means that the larger the quantity, the faster it decreases. The function \( e^{-0.05 t} \) in the algae accumulation equation is an example of exponential decay. As time \( t \) increases, this term becomes smaller and smaller because the exponent \(-0.05 t\) is negative, leading to a decay.

• Exponential decay is characterized by its rapid initial decline.
• It approaches zero but never actually reaches it.
• This type of decay has applications in various fields like physics, finance, and, as in our exercise, environmental science.

The significance of exponential decay in this context is that it reflects the diminishing availability of resources, such as oxygen in the lake. Over time, despite any oscillating factors, the algae growth rate will be dominated by this decay, ultimately trending towards zero.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. A classic example is the sine function, represented in our equation by \( \sin^2(\pi t) \). This particular term arises from the natural cycles, such as day and night.

• Periodic functions have cycles that repeat at regular intervals.
• They can model phenomena that exhibit regular patterns over time, like seasons or tides.
• The period of \( \sin^2(\pi t) \) corresponds to the time taken for one complete cycle.

In the algae equation, the periodic function reflects the diurnal (daily) changes in environmental factors, like sunlight and temperatures, contributing to the fluctuations in algae growth rates. However, this amplitude of change remains bounded due to its oscillating nature.

Oscillation

Oscillation refers to the repeated variation around a central value or between two or more different states. \( \sin^2(\pi t) \) in the equation represents oscillation in the algae growth.

• Oscillations are characterized by their swings back and forth.
• They often represent repetitive fluctuations seen in both natural and engineered systems.
• In this case, the oscillation is around the daily cycle, or diurnal changes.

While oscillation itself does not lead to increased growth in algae over an infinite time, it does introduce variability. This is important in understanding the cyclic nature of environmental systems, but when coupled with exponential decay, it ensures that long-term growth does not remain unchecked or become infinite. It's the balance of oscillation with decay that leads to the eventual stabilization or reduction of the algae population in the lake.