Problem 20 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

In two bodies of water, \(L_{1}\) and \(L_{2},\) the light intensities \(I_{1}(x)\) and \(I_{2}(x)\) as functions of depth \(x\) are measured simultaneously and found to be $$I_{1}(x)=800 e^{-0.04 x} \quad \text { and } \quad I_{2}(x)=700 e^{-0.05 x}$$ Explain the differences in the two formulas in terms of the properties of water in the two bodies.

Step-by-Step Solution

Answer

\(L_1\) has a higher surface light intensity and slower decay rate, indicating clearer water compared to \(L_2\).

Step 1: Identify common components

Both light intensity formulas are in the form of exponential decay functions. They show how light intensity decreases with increasing depth in each water body: \( I_{1}(x) = 800 e^{-0.04 x} \) and \( I_{2}(x) = 700 e^{-0.05 x} \).

Step 2: Compare initial intensities

The initial intensity at the surface (\(x=0\)) for \(L_1\) is 800 units, while for \(L_2\) it is 700 units. This suggests that water body \(L_1\) has more light penetration at the surface compared to \(L_2\).

Step 3: Analyze the decay rates

In the formulas, \(-0.04\) and \(-0.05\) are the decay rates. For water body \(L_1\), the decay rate \(0.04\) is smaller than in \(L_2\), where it is \(0.05\). This indicates that light intensity decreases more slowly in \(L_1\), implying \(L_1\) is less absorptive or clearer than \(L_2\).

Step 4: Describe overall light absorption

Given the differences in decay rates, \(L_1\) absorbs less light per unit depth than \(L_2\). This means that light penetrates deeper into \(L_1\) than \(L_2\), suggesting different clarity or turbidity levels in the two water bodies.

Key Concepts

Light Intensity

Light intensity refers to the amount of light that penetrates a medium, such as water. It is an important parameter in environmental science, particularly in understanding aquatic ecosystems.

In our exercise, we use exponential decay functions to represent how light diminishes with depth in water bodies. These functions, namely \( I_{1}(x) = 800 e^{-0.04 x} \) and \( I_{2}(x) = 700 e^{-0.05 x} \), describe the intensity of light at a point under the water surface.

• At the water surface (\( x=0 \)), the light intensities are highest.
• The constants 800 and 700 show the initial intensity values right at the surface for each water body.
• As depth (\( x \)) increases, the amount of light diminishes exponentially due to absorption and scattering.

Understanding light intensity in water helps to determine how much light reaches various depths, affecting aquatic plants and animals that rely on photosynthesis.

Depth Measurement

When studying underwater environments, measuring the depth at which light penetrates water provides crucial insights into ecological dynamics.

Depth measurement in the given context is an independent variable, represented by \( x \) in our exponential formulas. As depth increases:

• Less light is able to reach deeper parts of the water due to absorption and scattering by particles and water molecules.
• This absorption is more pronounced in turbid water where more particles are present.

By analyzing how rapidly light diminishes with depth, scientists can infer physical and chemical properties of the water body, such as turbidity and clarity.

Turbidity

Turbidity measures how cloudy or clear water appears, influenced by particles suspended in it. It's a key factor affecting light intensity at varying depths.

Higher turbidity means more particles in water:

• These particles scatter light, reducing its intensity at any given depth.
• Turbid water absorbs more light, causing light to decay more rapidly in the water column.

In the exercise, water body \(L_2\) with a decay rate of \(-0.05\) might exhibit higher turbidity, indicating quick light absorption and scattering. Conversely, \(L_1\)'s lower rate of \(-0.04\) implies the water allows more light to penetrate, suggesting lower turbidity compared to \(L_2\).

Monitoring turbidity is essential to understand water quality, affecting aquatic plant growth and the behavior of organisms that rely on light.

Water Clarity

Water clarity refers to how clear water is, and how far light can penetrate in it. This concept is intricately linked with both turbidity and light intensity.

Clear water allows for deeper light penetration, promoting better conditions for aquatic life:

• A clear body of water typically exhibits a slower rate of light decay, as seen in \(L_1\) where light decay is \(-0.04\).
• In contrast, murkier water like \(L_2\), where decay is faster at \(-0.05\), allows less light to penetrate deep.

High water clarity is usually indicative of lower concentrations of suspended particles, meaning organisms that depend on sunlight have better chances to thrive.

By assessing factors like light intensity and turbidity, scientists can determine the clarity of the water, providing important information about the ecosystem's health.