Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Thickness of the ozone 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 \(k \approx 1.88, I_{0} / I\) 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: Analyze the given formula

The formula provided is \( \ln I_{0} - \ln I = kx \sec \theta \). We are asked to find the thickness \( x \) of the ozone layer. Begin by identifying variables: \( \ln(I_{0}/I) = 1.72 \), \( k = 1.88 \), and \( \theta = 12^\circ \). Convert \( I_{0}/I \) into the logarithmic expression using natural log: \( \ln(I_{0}/I) \approx \ln(1.72) \).

Step 2: Simplify the logarithmic expression

Calculate \( \ln(1.72) \) using a calculator. This gives \( \ln(1.72) \approx 0.5421 \). Replace the expression \( \ln I_{0} - \ln I \) with \( 0.5421 \). Now the formula is \( 0.5421 = 1.88 \times x \times \sec(12^\circ) \).

Step 3: Calculate secant of the angle

\( \sec \theta = \frac{1}{\cos \theta} \). Calculate \( \cos(12^\circ) \) using a calculator, which is approximately \( 0.9781 \). Hence, \( \sec(12^\circ) = \frac{1}{0.9781} \approx 1.0224 \).

Step 4: Solve for the thickness \( x \)

Substitute the values of \( k \), \( \sec \theta \), and \( \ln(I_{0}/I) \) into the equation: \( 0.5421 = 1.88 \times x \times 1.0224 \). Simplify the equation as follows: \( 0.5421 = 1.92208 \times x \). Solve for \( x \) by dividing both sides by \( 1.92208 \): \( x = \frac{0.5421}{1.92208} \approx 0.2821 \).

Step 5: Round the result

Round \( x \approx 0.2821 \) to the nearest 0.01 centimeter. Therefore, the thickness of the ozone layer is approximately \( 0.28 \) cm.

Key Concepts

Logarithmic Functions

Logarithmic functions often appear in scientific calculations, especially when dealing with exponential relationships or ratios such as light intensities. The formula from the exercise uses natural logarithms, which are logs with base "e", to simplify the ratio between initial and final light intensities, expressed as \( \ln(I_{0}/I) \). Natural logarithms help manage large differences in intensity values by scaling them down.
Specifically, in our case, \( I_{0} \) and \( I \) are the initial and final intensities of sunlight passing through the ozone layer. The expression \( \ln(I_{0}/I) \) essentially tells us how many times more intense the initial light is compared to the final light after it has been filtered by the ozone. This logarithmic approach is crucial because it simplifies multiplicative relationships into additive ones, which are much easier to handle mathematically.

Trigonometric Functions

In the context of this problem, trigonometric functions help account for the angle at which sunlight enters the atmosphere. Specifically, the secant function \( \sec(\theta) \) is used, which is the reciprocal of the cosine function, expressed as \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
When sunlight enters Earth's atmosphere at an angle, it travels through a longer path compared to when it comes in directly overhead. This is why the angle \( \theta \) affects the thickness estimation of the ozone layer. The secant function compensates for the angle by adjusting the measurement according to the path length. For example, when the sun is low in the sky (larger angle \( \theta \)), \( \sec(\theta) \) becomes larger, indicating an increased path length through the atmosphere.

Scientific Measurement

Scientific measurement is about precision and accuracy, especially when dealing with small or large quantities. The exercise requires precise computations with constants, angles, and measurements.
The wavelength of light given as \( 3055 \times 10^{-8} \) centimeters showcases scientific notation, essential for expressing very small measurements common in scientific contexts. It's a more practical way to handle these figures without being overwhelmed by the zeros.
Moreover, rounding measurements to specified precision—here, to the nearest 0.01 cm—is a standard practice in scientific reporting. This precision ensures reliability in data and results, while also acknowledging the inherent limitations of measurement instruments and calculations. By rounding, we align our results with realistic accuracy achievable with the measurement tools available.