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Chapter 7: Problem 27

The intensity \(L(x)\) of light \(x\) feet 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 \(18 \mathrm{ft}\) 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?

Short Answer

Expert verified
The diver can work up to approximately 60 feet without artificial light.

Step by step solution

01

Write the Differential Equation

The differential equation given is \( \frac{dL}{dx} = -kL \). This implies that the rate of change of light intensity with respect to depth is proportional to the intensity itself, with \(-k\) as the constant of proportionality.
02

Solve the Differential Equation

This is a separable differential equation. Rewriting it, we have \( \frac{dL}{L} = -k \, dx \). Integrating both sides yields \( \int \frac{dL}{L} = \int -k \, dx \), which gives \( \ln |L| = -kx + C \). Exponentiating both sides, we get the general solution \( L(x) = C e^{-kx} \).
03

Apply Initial Conditions to Find Constants

We are given that at 18 feet, the light intensity is half of its surface value. Thus, \( L(18) = \frac{1}{2} L(0) \). Substituting into the general solution, \( \frac{1}{2} C = C e^{-18k} \). This simplifies to \( e^{-18k} = \frac{1}{2} \). Solving for \( k \), we have \( k = \frac{\ln 2}{18} \).
04

Determine Depth Without Artificial Light

The diver can work without artificial light until the intensity is more than one-tenth of the surface intensity. Thus, \( L(x) = \frac{1}{10} L(0) \). Substituting in the expression for \( L(x) \), \( \frac{1}{10} C = C e^{-kx} \). Simplifying, we have \( e^{-kx} = \frac{1}{10} \). Substituting the value of \( k \), we get \( -\frac{\ln 2}{18} x = \ln\left(\frac{1}{10}\right) \). Solving for \( x \), \( x = \frac{18 \ln 10}{\ln 2} \).
05

Calculate the Maximum Depth

Substitute values to find the numerical depth. Using \( \ln 10 \approx 2.302 \) and \( \ln 2 \approx 0.693 \), we calculate \( x \approx \frac{18 \times 2.302}{0.693} \). Therefore, the maximum depth is approximately 60 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Light Intensity
Light intensity refers to the amount of light energy falling on a surface. In the context of this exercise, it signifies how much light reaches different depths under the ocean's surface. As we go deeper into the ocean, the light intensity typically decreases exponentially. This happens because water absorbs and scatters light, reducing its intensity as it penetrates deeper. Divers often use light intensity as a crucial factor to determine how deep they can operate without additional light sources. It's essential to know that at certain depths, the light intensity may fall too low, hindering visibility underwater. To quantify this, mathematicians use the concept of differential equations, which is helpful in modeling scenarios where light intensity changes continuously with depth.
Exponential Decay
Exponential decay is a mathematical concept where quantities decrease over time at a rate proportional to their current value. In our problem, the light intensity decreases exponentially as the diver moves deeper into the ocean. This means the rate of decrease of light intensity is proportional to its current value. The equation governing exponential decay can be expressed as:
  • \( \frac{dL}{dx} = -kL \)
Here, \( L \) stands for the light intensity, \( x \) is the depth, and the negative sign indicates a decrease. Exponential decay is a common phenomenon in nature and describes many processes where quantities reduce over a given factor repeatedly over time or space. By understanding exponential decay, divers can predict how quickly light diminishes as they descend, enabling better planning and safety precautions for diving activities.
Separable Differential Equation
A separable differential equation is a type of differential equation in which the variables can be separated maneuverably on each side of the equation for an easy integration process. This characteristic makes it more manageable to solve and understand. In our problem:
  • The differential equation \( \frac{dL}{dx} = -kL \) is separable because the variables can be arranged as \( \frac{dL}{L} = -k \, dx \).
  • Both sides can be integrated independently to solve for the general solution.
Solving through integration gives:
  • \( \int \frac{dL}{L} = \int -k \, dx \)
  • Resulting in: \( \ln |L| = -kx + C \)
Exponentiating both sides, we get the function:
  • \( L(x) = Ce^{-kx} \)
This solved equation allows us to evaluate the light intensity at any given depth, providing divers with the necessary information to anticipate the lighting conditions they might face underwater.

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Most popular questions from this chapter

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. \(2 y^{\prime}+3 y=e^{-x}\) a. \(y=e^{-x}\) b. \(y=e^{-x}+e^{-(3 / 2) x}\) c. \(y=e^{-x}+C e^{-(3 / 2) x}\)

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