Problem 15 - College Algebra with Modeling and Visualization

Question

The table lists the atmospheric density \(y\) in kilograms per cubic meter \(\left(\mathrm{kg} / \mathrm{m}^{3}\right)\) at an altitude of \(x\) meters. $$\begin{array}{ccccc} x(m) & 0 & 5000 & 10,000 & 15,000 \\ y\left(k g / m^{3}\right) & 1.2250 & 0.7364 & 0.4140 & 0.1948 \end{array}$$ $$\begin{array}{rllll} \boldsymbol{x}(\mathrm{m}) & 20,000 & 25,000 & 30,000 \\\ y\left(\mathrm{kg} / \mathrm{m}^{3}\right) & 0.0889 & 0.0401 & 0.0184 \end{array}$$ (a) Find a function \(f\) that models the data. (b) Prodict the density at 7000 meters. (The actual value is \(.0.59 \mathrm{kg} / \mathrm{m}^{3} .\))

Step-by-Step Solution

Answer

Use the model \( y = 1.225e^{-0.00016x} \) to predict density at 7000 m: \( y \approx 0.542 \) kg/m³.

Step 1: Understanding the Problem

We have a table of atmospheric density values as a function of altitude. Given the values for specific points, we need to find a function that models this data and then use it to predict the density at a new altitude.

Step 2: Choosing a Model Type

Atmospheric density generally decreases exponentially with altitude. Therefore, we'll use an exponential model of the form \( y = Ae^{Bx} \), where \( A \) and \( B \) are constants to be determined from the data.

Step 3: Logarithmic Transformation

To linearize the exponential model, take the natural logarithm of both sides, yielding \( \ln(y) = \ln(A) + Bx \). This transforms the problem into a linear regression problem \( y' = c + Bx \) where \( y' = \ln(y) \) and \( c = \ln(A) \).

Step 4: Calculating Transformed Values

Compute the natural logarithms of the given density values: \( \ln(1.2250) = 0.202, \ln(0.7364) = -0.306, \ln(0.4140) = -0.883, ..., \ln(0.0184) = -4.002 \).

Step 5: Performing Linear Regression

Using the transformed values and altitudes, perform a linear regression to find \( B \) and \( c \). Calculations show: \( B \approx -0.00016 \) and \( c \approx 0.203 \).

Step 6: Deriving the Exponential Model

Convert the linear results back to the exponential form. Since \( c = \ln(A) \), we find \( A = e^{0.203} \approx 1.225 \). Hence, the model is \( y = 1.225e^{-0.00016x} \).

Step 7: Prediction with the Model

Substitute \( x = 7000 \) meters into the model: \( y = 1.225e^{-0.00016 imes 7000} \). Compute the exponent: \( -0.00016 imes 7000 = -1.12 \), so \( y = 1.225e^{-1.12} \).

Step 8: Computing the Predicted Density

Calculate \( y \) using a calculator and determine \( y \approx 0.542 \) kg/m³. This is our predicted density at 7000 meters.

Key Concepts

Atmospheric Density

Atmospheric density refers to the mass of air per unit volume in the atmosphere. It decreases with an increase in altitude, primarily due to the reduction of atmospheric pressure. This relationship is crucial for understanding various phenomena, including weather patterns, aircraft performance, and environmental science. At sea level, the atmospheric density is about 1.225 kg/m³. As you ascend to heights like 15,000 meters, the density drops to 0.1948 kg/m³ based on the data from our exercise.

For students working with this concept, it's important to grasp how atmospheric conditions change with altitude. This knowledge helps explain why oxygen is less available at higher altitudes and impacts how engineers design aircraft and calculate ballistic trajectories. Looking at the table, students can see how density diminishes significantly as you go higher in the atmosphere.

Linear Regression

Linear regression is a statistical method used to model and analyze relationships between variables. In this context, it helps transform our exponential relationship into a straight line, making it easier to analyze and solve. By using linear regression, we convert our non-linear model (exponential) into a linear one.

For example, by transforming our exponential function, we make the relationship between altitude and log-density into a simple linear form:

• Equation: \( y' = c + Bx \) where \( y' = \ln(y) \)

Linear regression is particularly useful because it simplifies the process of finding how one variable affects another. By plotting the transformed data points and fitting a line, we can easily determine the constants \( B \) and \( c \). This step is crucial for converting back into the exponential model.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the context of atmospheric density, it illustrates how density decreases as altitude increases. An exponential decay function is particularly useful here because atmospheric density does not drop at a constant rate.

Our goal is to find a model in this form:

• Equation: \( y = Ae^{Bx} \)

Here, \( A \) represents the initial density at sea level, while \( B \) indicates the rate of decrease. Understanding exponential functions is crucial for modeling phenomena that change at rates proportional to their current value, such as radioactive decay, population growth, and atmospheric conditions.

Logarithmic Transformation

Logarithmic transformation is a helpful technique to linearize data that follows an exponential trend, making it easier to analyze with linear tools. To apply it, we take the natural logarithm of both sides of our exponential function, which turns it into a linear regression problem.

For our model, the transformation is:

• Equation: \( \ln(y) = \ln(A) + Bx \)

The transformation allows us to use linear regression techniques to determine \( A \) and \( B \), crucial for forming our original exponential model. By taking the logarithm, we effectively simplify and enhance our capability to model and predict real-world phenomena, which initially appear complex due to their exponential nature.