Problem 39 - Precalculus: Functions and Graphs

Question

Thichness of the azone layer The thickness of the ozone layer can be estimated using the formula $$ \ln I_{0}-\ln I=k x \sec \theta $$ where \(I_{0}\) is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, \(I\) is the intensity of the same wavelength after passing through a layer of ozone \(x\) centimeters thick, \(k\) is the absorption constant of ozone for that wavelength, and \(\theta\) is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of \(3055 \times 10^{-8}\) centimeter with \(k=1.88, I_{0} / I\) is measured as 1.72 and \(\theta=12^{\circ} .\) Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

Step-by-Step Solution

Answer

The thickness of the ozone layer is approximately 0.28 cm.

Step 1: Understand the Equation

The given equation is \( \ln I_{0} - \ln I = k x \sec \theta \). This can be rewritten as \( \ln\left(\frac{I_{0}}{I}\right) = k x \sec \theta \), because the difference of logarithms is the logarithm of the quotient. Our goal is to find the thickness \( x \) of the ozone layer.

Step 2: Substitute Known Values

You are given \( \frac{I_{0}}{I} = 1.72 \), \( k = 1.88 \), and \( \theta = 12^{\circ} \). Substitute these values into the equation: \( \ln(1.72) = 1.88 \times x \times \sec(12^{\circ}) \).

Step 3: Calculate \(\ln(1.72)\)

Compute the natural logarithm of 1.72 using a calculator. \( \ln(1.72) \approx 0.541 \).

Step 4: Calculate \(\sec(12^{\circ})\)

The secant is the reciprocal of cosine. First, find \( \cos(12^{\circ}) \), then \( \sec(12^{\circ}) = \frac{1}{\cos(12^{\circ})} \). Calculate this using a calculator: \( \cos(12^{\circ}) \approx 0.978 \), so \( \sec(12^{\circ}) \approx 1.023 \).

Step 5: Solve for \(x\)

Substitute \( \ln(1.72) \) and \( \sec(12^{\circ}) \) back into the equation. \( 0.541 = 1.88 \times x \times 1.023 \). Rearrange and solve for \( x \): \[ x = \frac{0.541}{1.88 \times 1.023} \].

Step 6: Calculate and Round \(x\)

Compute \( x \) using a calculator. \( x \approx \frac{0.541}{1.92264} \approx 0.2814 \). Round this result to the nearest 0.01 centimeter, which gives \( x \approx 0.28 \) cm.

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms. In our context, we use them to solve problems related to the intensity of light. A logarithm helps in determining how many times a number must be multiplied by itself to arrive at another number. This is handy when handling wide-ranging values, like the intensities of light rays.
First, let's revisit the formula: \( \ln I_{0} - \ln I = k x \sec \theta \). This can be expressed as \( \ln \left( \frac{I_{0}}{I} \right) = k x \sec \theta \).

• \( \ln \left( \frac{I_{0}}{I} \right) \) indicates the logarithm of the ratio of initial and final light intensities.
• The subtraction of logarithms simplifies to a division within the logarithm.

Understanding logarithmic equations helps us to manipulate them for solving unknown variables efficiently.

Secant Function

The secant function, abbreviated as \( \sec \theta \), is a trigonometric function related to the cosine. For any angle \( \theta \), secant is defined as the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \). It is crucial in relating angles to the intensity of sunlight.
When sunlight strikes the atmosphere, the angle \( \theta \) helps determine the path length of light through the ozone. Here, we used \( \theta = 12^{\circ} \), which means:

• Calculate \( \cos(12^{\circ}) \), which is roughly 0.978.
• The secant function is then \( \sec(12^{\circ}) \approx 1.023 \).

Understanding the secant function allows us to effectively model how sunlight impacts the ozone layer, influencing our calculation of ozone thickness.

Absorption Constant

The absorption constant, denoted as \( k \), is a measurement that describes how a material absorbs light. Each material has a unique absorption constant based on its properties and the specific wavelength of light.
In our given formula, \( k = 1.88 \) for the specific wavelength mentioned.

• This constant helps us quantify the amount of light absorbed by the ozone.
• A higher \( k \) reflects more absorption, indicating a denser or more effective layer of ozone in blocking that wavelength of light.

Having a precise absorption constant is essential for calculating the thickness of atmospheric layers like the ozone.

Wavelength Measurement

Wavelength is the distance between repeating units of a wave, and in this equation, it is crucial for understanding how different wavelengths of light interact with the ozone layer.
Our problem deals with a wavelength of \( 3055 \times 10^{-8} \) centimeters. This measurement is important because:

• Different wavelengths are absorbed differently by the atmosphere.
• The specified wavelength determines the absorption behavior modeled by the constant \( k \).

Understanding wavelength measurements and how specific wavelengths interact with layers like the ozone helps us accurately predict how much light is absorbed before reaching the Earth's surface.