Problem 85 - College Algebra with Modeling and Visualization

Question

Light Absorption When sunlight passes through lake water, its initial intensity \(I_{0}\) decreases to a weaker intensity \(I\) at a depth of \(x\) feet according to the formula $$ \ln I-\ln I_{0}=-k x $$ where \(k\) is a positive constant. Solve this equation for \(I .\) (PICTURE NOT COPY)

Step-by-Step Solution

Answer

The intensity \( I \) at depth \( x \) is given by \( I = I_{0} e^{-kx} \).

Step 1: Understand the Equation

Recognize the given equation: \( \ln I - \ln I_{0} = -k x \). This equation describes the change in intensity of light as it passes through water, where \( I \) is the intensity at depth \( x \), \( I_{0} \) is the initial intensity at the surface, and \( k \) is a constant representing absorption.

Step 2: Use Logarithmic Properties

Use the property of logarithms that states \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). Apply this to the equation: \( \ln \left(\frac{I}{I_{0}}\right) = -kx \).

Step 3: Remove the Logarithm

Exponentiate both sides to remove the natural logarithm. This gives: \( \frac{I}{I_{0}} = e^{-kx} \).

Step 4: Solve for I

Multiply both sides of the equation by \( I_{0} \) to solve for \( I \): \( I = I_{0} e^{-kx} \).

Key Concepts

Light Absorption

Light absorption is a fascinating natural process that describes how the intensity of light changes as it travels through a medium like water. Imagine sunlight entering a lake: as it moves deeper, less light is available. The change in light intensity, or how much light gets weaker, is described by an equation with a constant called the absorption coefficient, represented by "k." This coefficient tells us how quickly or slowly light becomes weaker based on the properties of the lake water.

• Higher values of "k" mean light is absorbed more quickly, reducing the intensity faster.
• Lower values of "k" indicate that the light maintains its intensity longer as it travels deeper.

Understanding light absorption is crucial in natural sciences as it influences how we perceive underwater environments and affects biological processes like photosynthesis in plants.

Logarithmic Properties

Logarithms are powerful mathematical tools that simplify complex multiplicative relationships by transforming them into manageable additive ones. One key property, particularly useful in solving equations involving exponential changes—like our light absorption problem—is that \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). This property helps us manage equations where quantities are compared as ratios.When we apply this property to problems like light absorption, we transform the relationship of intensities from subtraction to division, making it easier to isolate variables. Logarithmic properties serve as a bridge between straightforward arithmetic and more complex exponential scenarios, helping us streamline processes and solve equations efficiently.It's important to feel comfortable using logarithmic properties, as they are applicable across different fields of study, from chemistry in pH calculations to economics in budgeting and investments.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another. It serves as the inverse process of taking a logarithm. In our exercise with light absorption, once we've applied the logarithmic property to simplify the equation, exponentiation allows us to "undo" the natural logarithm and solve for our desired variable.For instance, in the equation \( \ln \left(\frac{I}{I_{0}}\right) = -kx \), exponentiating both sides removes the natural logarithm, resulting in \( \frac{I}{I_{0}} = e^{-kx} \). This step is critical as it converts the problem from a logarithmic form back into an exponential form, revealing a clear exponential decay relationship.Exponentiation is especially significant in fields dealing with exponential growth and decay, such as population modeling, finance, and radioactive decay.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm with a base of \( e \), where \( e \approx 2.71828 \). It is prominent in mathematical equations that model natural phenomena, such as the exponential decay observed in light absorption.The natural logarithm "ln" simplifies the calculation of changes over continuous intervals and is preferred in calculus and engineering fields. The property \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) takes full advantage of the natural logarithm's characteristics, making the manipulation of physical and theoretical data more practical.Understanding how natural logarithms operate is essential for students and professionals alike, as they appear frequently in calculations involving continuous growth processes, like interest rates and population dynamics, besides natural decay processes like that of light absorption.