Problem 19 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose solar radiation striking the ocean surface is \(1250 \mathrm{~W} / \mathrm{m}^{2}\) and 20 percent of that energy is reflected by the surface of the ocean. Suppose also that 20 meters below the surface the light intensity is found to be \(800 \mathrm{~W} / \mathrm{m}^{2}\). a. Write an equation descriptive of the light intensity as a function of depth in the ocean. b. Suppose a coral species requires \(100 \mathrm{~W} / \mathrm{m}^{2}\) light intensity to grow. What is the maximum depth at which that species might be found?

Step-by-Step Solution

Answer

The equation is: \(I(d) = 1000 \times e^{-0.0111d}\). The maximum depth for coral growth is approximately 208 meters.

Step 1: Calculate Net Solar Radiation

First, we need to determine the net solar radiation that is not reflected by the surface. Given that 20% of the solar radiation is reflected, the remaining 80% is absorbed. Thus, the net solar radiation is computed as:\[I_0 = 1250 \times (1 - 0.20) = 1250 \times 0.80 = 1000 \ \text{W/m}^2\]where \(I_0\) is the initial intensity after reflection.

Step 2: Define the Intensity as a Function of Depth

The intensity of light in water decreases exponentially with depth due to absorption. We can model this with the equation:\[I(d) = I_0 \times e^{-kd}\]where \(I(d)\) is the intensity at depth \(d\), \(I_0\) is the initial intensity calculated in step 1, and \(k\) is the attenuation coefficient.

Step 3: Determine the Attenuation Coefficient

We use the given data that the intensity is 800 W/m² at 20 meters below the surface to find \(k\). Substitute the known values into the intensity function:\[800 = 1000 \times e^{-20k}\]Solving for \(k\), we get:1. Divide both sides by 1000: \[0.8 = e^{-20k}\]2. Take the natural logarithm of both sides:\[\ln(0.8) = -20k\]3. Solve for \(k\):\[ k = -\frac{\ln(0.8)}{20} \approx 0.0111 \ \text{m}^{-1}\]

Step 4: Write the Full Light Intensity Equation

Now, substitute \(I_0\) and \(k\) back into the originally defined function:\[I(d) = 1000 \times e^{-0.0111d}\]This is the equation describing light intensity as a function of depth in the ocean.

Step 5: Calculate the Maximum Depth for Coral Growth

To find the maximum depth where the light intensity is at least 100 W/m² for coral growth, set \(I(d) = 100\) and solve for \(d\):\[100 = 1000 \times e^{-0.0111d}\]1. Divide both sides by 1000:\[0.1 = e^{-0.0111d}\]2. Take the natural logarithm of both sides:\[\ln(0.1) = -0.0111d\]3. Solve for \(d\):\[ d = -\frac{\ln(0.1)}{0.0111} \approx 207.94 \ \text{meters}\]Therefore, the coral species can grow up to approximately 208 meters depth.

Key Concepts

Exponential Decay

Exponential decay is a concept used to model various natural phenomena where a quantity decreases at a rate proportional to its current value. In the context of light intensity in water, exponential decay explains why light diminishes as it penetrates deeper. The key equation is \[ I(d) = I_0 \times e^{-kd} \]Here,

• \(I(d)\) represents the light intensity at depth \(d\).
• \(I_0\) is the initial light intensity just below the surface.
• \(k\) is the attenuation coefficient, representing how quickly the light fades.

This model helps us understand and predict how much light reaches various depths in the ocean, crucial for studying marine ecosystems.

Solar Radiation

Solar radiation refers to the energy emitted by the sun and is a primary energy source for Earth’s climate system. When solar radiation reaches the ocean, several factors influence its distribution:

• Reflection: Not all incoming solar radiation is absorbed; a percentage is reflected back into the atmosphere. In our exercise, 20% of solar radiation is reflected.
• Absorption: The remaining 80% of the solar energy penetrates and is absorbed by the ocean, contributing to warming surface waters and affecting marine life.

Understanding how solar radiation interacts with the ocean surface is vital for calculating light availability at various depths, affecting photosynthetic organisms and temperature regulation in marine environments.

Attenuation Coefficient

The attenuation coefficient \(k\) quantifies how quickly light is absorbed as it travels through a medium like water. In simpler terms, it measures the rate of light decrease. The attenuation coefficient is determined through observations, as shown in the exercise, where data revealed light intensity at specific depths:- At the ocean's surface, initial light intensity (after reflection) is known.- At 20 meters below, the observed intensity allows us to calculate \(k\).This coefficient is essential for predicting how deeply sunlight can penetrate and supports ecological studies by indicating where photosynthesis is possible.

Ocean Depth Analysis

Analyzing ocean depths involves understanding how light diminishes at various levels below the surface. This study is important for marine biology, especially in areas like coral growth:

• Corals, which require a minimum light intensity (e.g., 100 W/m²) for photosynthesis, depend on these depth calculations to thrive.
• Equation-derived values provide insights, helping predict the maximum depth at which certain life forms can exist.

Using the light intensity equation, we calculate such depths crucial for ecosystems. For instance, the problem demonstrates that corals might grow up to nearly 208 meters deep, offering valuable data for conservation and marine research.