Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Light penetration in an ocean An important problem in oceanography is to determine the amount of light that can penetrate to various ocean depths. The Beer-Lambert law asserts that the exponential function given by \(I(x)=I_{0} c^{x}\) is a model for this phenomenon (see the figure). For a certain location, \(I(x)=10(0.4)^{x}\) is the amount of light (in calories \(\left./ \mathrm{cm}^{2} / \mathrm{sec}\right)\) reaching a depth of \(x\) meters. (a) Find the amount of light at a depth of 2 meters. (b) Sketch the graph of \(I\) for \(0 \leq x \leq 5\).

Step-by-Step Solution

Answer

(a) 1.6 calories/cm²/sec at 2 meters depth. (b) Plot decreases exponentially from 10 to 0.1024 over 0 to 5 meters.

Step 1: Understand the Problem

We are given a function, based on Beer-Lambert's law, that models the amount of light penetrating the ocean at a depth \(x\) meters. The function is \(I(x) = 10(0.4)^x\), where \(I_0 = 10\) and the constant \(c = 0.4\). We need to evaluate this function at \(x = 2\) and sketch it for \(x\) ranging from 0 to 5.

Step 2: Evaluate the Function at x = 2

Substitute \(x = 2\) into the function and calculate: \[I(2) = 10 \times (0.4)^2 = 10 \times 0.16 = 1.6\]. This means the light intensity at 2 meters depth is 1.6 calories/cm²/sec.

Step 3: Sketch the Function

To sketch the function, identify key points by evaluating \(I(x)\) at integer values of \(x\) from 0 to 5. - When \(x = 0\), \(I(0) = 10 \times (0.4)^0 = 10\) - When \(x = 1\), \(I(1) = 10 \times 0.4 = 4\)- We have already found \(I(2) = 1.6\)- When \(x = 3\), \(I(3) = 10 \times (0.4)^3 = 0.64\)- When \(x = 4\), \(I(4) = 10 \times (0.4)^4 = 0.256\)- When \(x = 5\), \(I(5) = 10 \times (0.4)^5 = 0.1024\)Plot these points on a graph with \(x\)-axis representing depth in meters, and \(y\)-axis representing light intensity.

Key Concepts

Exponential Function

An exponential function is a mathematical expression written in the form \(f(x) = a \, b^x\), where \(a\) is a constant and \(b\) is the base of the exponential. This function shows how quantities grow or decay exponentially over time. It is characterized by its rapid rate of change, either increasing or decreasing.

Some key characteristics of exponential functions include:

• The base \(b\) determines the nature of the function. If \(b > 1\), the function models exponential growth. If \(0 < b < 1\), it represents exponential decay.
• The initial value \(a\) is the starting point of the function when \(x = 0\).
• Exponential functions are not symmetric.

In the case of light penetration modeled by the Beer-Lambert law, the given function \(I(x) = 10(0.4)^x\) is an example of exponential decay, as the base \(0.4\) is less than one, causing the light intensity to decrease as the depth \(x\) increases.
This decreasing function accurately reflects how light diminishes as it travels deeper into the ocean.

Light Penetration

Light penetration in bodies of water is crucial in oceanography. It determines the amount of sunlight that reaches different depths in the ocean, influencing marine ecosystems.

Here’s what you should know about light penetration:

• Visible light, essential for processes like photosynthesis, is progressively absorbed and scattered as it moves through water.
• The rate at which light diminishes depends on factors such as water clarity, the presence of particles, and dissolved substances.
• The Beer-Lambert law mathematically models this decline in light intensity with depth in the ocean.

The Beer-Lambert law, using an exponential function, shows that light intensity \(I(x)\) declines exponentially with depth \(x\). The equation \(I(x)=10(0.4)^{x}\) indicates this decline through an exponential decay function, where \(10\) is the original intensity (at the surface), and \(0.4\) specifies how rapidly the intensity falls. This model helps scientists estimate how much light is available at different depths, vital for marine biology research and environmental monitoring.

Oceanography

Oceanography is the scientific study of oceans, encompassing aspects such as marine ecosystems, currents, waves, and ocean floor geology. In understanding light penetration, oceanographers are interested in how varying factors affect the amount of sunlight reaching underwater environments, influencing biological and chemical processes.

Key components of oceanography include:

• The study of physical and chemical properties of ocean waters.
• Understanding ocean currents and their effects on climate and weather patterns.
• Investigating marine life and ecosystems and how organisms adapt to their environment.

Light penetration is of particular interest as it affects photosynthesis and determines the distribution of marine life in the photic zone, where sufficient sunlight supports life. Technologies like remote sensing and underwater sensors help gather data on light absorption and scattering. The application of mathematical models, such as the Beer-Lambert law, further aids oceanographers in predicting light behavior and its implications on the ocean ecosystem.