Problem 27 - Thomas Calculus in SI Units

Question

The intensity \(L(x)\) of light \(x\) meters beneath the surface of the ocean satisfies the differential equation $$\frac{d L}{d x}=-k L$$ As a diver, you know from experience that diving to 6 meters in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light?

Step-by-Step Solution

Answer

You can work without artificial light up to about 19.9 meters deep.

Step 1: Understand the Differential Equation

The given differential equation is \( \frac{dL}{dx} = -kL \), which describes the rate of change of light intensity with respect to depth \( x \). It indicates that the intensity decreases exponentially with depth.

Step 2: Find the Solution to the Differential Equation

The general solution of the differential equation \( \frac{dL}{dx} = -kL \) is an exponential function of the form \( L(x) = L_0 e^{-kx} \), where \( L_0 \) is the intensity at the surface (i.e., when \( x = 0 \)).

Step 3: Apply the Given Condition

We know that diving to 6 meters cuts the intensity in half, so \( L(6) = \frac{L_0}{2} \). Substituting into the solution, we have \( \frac{L_0}{2} = L_0 e^{-6k} \). Dividing both sides by \( L_0 \) gives \( \frac{1}{2} = e^{-6k} \).

Step 4: Solve for the Constant k

Take the natural logarithm of both sides of \( \frac{1}{2} = e^{-6k} \): \( \ln\left(\frac{1}{2}\right) = -6k \). Solving for \( k \), we get \( k = -\frac{\ln\left(\frac{1}{2}\right)}{6} = \frac{\ln(2)}{6} \).

Step 5: Determine Safe Depth Without Artificial Light

You cannot work when the intensity falls below one-tenth of the surface value, so \( L(d) = \frac{L_0}{10} \). Using the solution formula \( \frac{L_0}{10} = L_0 e^{-kd} \), cancel \( L_0 \) to get \( \frac{1}{10} = e^{-kd} \).

Step 6: Solve for Depth d

Taking the natural logarithm, \( \ln\left(\frac{1}{10}\right) = -kd \). Substitute \( k = \frac{\ln(2)}{6} \) to get \( \ln\left(\frac{1}{10}\right) = -\frac{\ln(2)}{6}d \). Solve for \( d \): \( d = \frac{6 \ln\left(\frac{1}{10}\right)}{-\ln(2)} \).

Step 7: Calculate the Numerical Value

Compute \( \ln\left(\frac{1}{10}\right) \approx -2.302 \) and \( \ln(2) \approx 0.693 \). Therefore, \( d \approx \frac{6(-2.302)}{-0.693} \approx 19.9 \). So you can work safely up to about 19.9 meters.

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept used to describe processes that decrease over time at a rate proportional to their current value. This concept is commonly used in various fields such as physics, biology, and finance.
In the context of differential equations, exponential decay is represented by the equation \( \frac{dL}{dx} = -kL \). Here, \( L \) is the light intensity, \( x \) is the depth, and \( k \) is a positive constant representing the rate of decay.
This equation tells us that as we go deeper into the ocean, the light intensity \( L \) decreases exponentially. This means the deeper the depth, the faster the light diminishes. The solution to this kind of equation is typically an exponential function, which regulates how quickly quantities decrease over time or depth.

Light Intensity

Light intensity refers to the amount of light that travels through a medium, in this case, water. In the ocean, light intensity greatly diminishes as you dive deeper. This phenomenon can be described mathematically using differential equations.
At the surface of the ocean, light intensity is at its maximum, typically referred to as \( L_0 \). As light penetrates the water, it starts to lose strength due to absorption and scattering by water molecules and other particles.
In this context, the initial condition given, where diving to 6 meters cuts the intensity by half, provides an important piece of information for predicting light levels at various depths. The challenge is to calculate how much light remains at different depths and determine the maximum depth a diver can reach before needing artificial light.

Depth Calculation

Calculating the depth to which a diver can go without artificial light involves using the mathematical model of exponential decay. Here, the solution \( L(x) = L_0 e^{-kx} \) represents light intensity at depth \( x \). To find the safe working depth, it's necessary to determine at what point this intensity reaches one-tenth of its surface value.
Given the condition that light is halved at 6 meters, we solve for the decay constant \( k \) using \( \ln\left(\frac{1}{2}\right) = -6k \). Our goal is to estimate when the intensity reduces to one-tenth, described by \( L(d) = \frac{L_0}{10} \).
Ultimately, the calculation for depth \( d \) involves using natural logarithms to solve \( d = \frac{6 \ln\left(\frac{1}{10}\right)}{-\ln(2)} \), translating to approximately 19.9 meters. At this depth, artificial light becomes necessary.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that gives the power to which the base \( e \) (approximately 2.718) must be raised to obtain a given number. It is essential in solving exponential equations, especially in the context of exponential decay problems.
In our light intensity problem, natural logarithms are used to manipulate and solve the exponential functions that describe how light decreases with depth. By applying \( \ln \) to both sides of an equation like \( \frac{1}{10} = e^{-kd} \), it transforms into a linear form which is easier to solve.
This conversion simplifies complex exponential relationships, allowing for precise calculations, such as determining the exact depth where artificial lighting becomes necessary when given an exponential decay of light intensity. Using natural logarithms ensures that we accurately capture how natural processes, like light absorption, behave.