Problem 134 - College Algebra Essentials

Question

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\mathrm{TRACE}]\) and \([\mathrm{ZOOM}]\) features or the intersect command of your graphing utility to verify your answer.

Step-by-Step Solution

Answer

The barometric air pressure is 29 inches of mercury approximately 63.47 miles from the eye of the hurricane.

Step 1: Understand the Problem

The problem provides a function \(f(x)\) which represents the barometric air pressure at a distance \(x\) from the eye of a hurricane. The problem asks how far from the eye of the hurricane the barometric air pressure is 29 inches of mercury. Given the function \(f(x)=0.48 \ln (x+1)+27\), we need to find \(x\) when \(f(x)=29\)

Step 2: Set up the Equation

Given that the air pressure is 29 inches of mercury, we set the function equal to 29: \(0.48 \ln(x+1) + 27 = 29\)

Step 3: Solve for \(x\)

We first isolate the \(0.48 \ln(x+1)\) term by subtracting 27 from both sides to obtain: \(0.48 \ln(x+1) = 2\). Then, we divide both sides by 0.48 to get: \(\ln(x+1) = \frac{2}{0.48}\) which simplifies to: \(\ln(x+1) = 4.16667\). Next, we raise both sides to the base of the natural logarithm (e) to get rid of the logarithm on the left side: \(x+1 = e^{4.16667}\). Finally, subtract 1 from both sides to solve for \(x: x = e^{4.16667} - 1\)

Step 4: Numerical approximation

To get an approximate value for the answer, calculate the value of \(e^{4.16667} - 1\) using an appropriate calculator to obtain: \(x = 63.47\) (rounded to 2 decimal places)

Step 5: Verify Solution

Use a calculator's TRACE or ZOOM feature or the intersect command on a graphing utility to verify the solution is correct.

Key Concepts

Natural Logarithm Applications

The natural logarithm is a cornerstone concept in mathematics, mainly because it is the inverse function of the exponential function. In real-world scenarios, like modeling the pressure changes within a hurricane, the natural logarithm helps us describe growth rates and decay processes.

For instance, when the barometric pressure changes as a function of distance from the eye of a hurricane are given by a logarithmic function, this indicates an exponential change in air pressure with respect to distance. The logarithmic scale transforms the exponential nature of this change into a linear one, making it easier to analyze and understand.

Furthermore, natural logarithms play a critical role in fields such as biology for growth and decay of populations, in finance for compound interest calculations, and in physics for decay of radioactive materials. Understanding how to apply the natural logarithm to solve real-life problems is an essential skill in many scientific and engineering disciplines.

Solving Exponential Equations

When we encounter an equation involving the natural logarithm, such as the one from our hurricane model, we often need to 'undo' the logarithmic function to solve for the variable of interest. This process typically involves converting the logarithmic equation into its equivalent exponential form.

Let's consider the step where we isolated the natural logarithm: \( \ln(x+1) = 4.16667 \). By exponentiating both sides with the base of \( e \), the base of the natural logarithms, we get \( x+1 = e^{4.16667} \). It's this delicate conversion which allows us to derive a solution that is otherwise wrapped in the complexity of a logarithmic relationship.

Understanding how to solve exponential equations is crucial not only in mathematics but also in analyzing phenomena involving growth and decay, such as population dynamics, radioactive decay, and, as presented in our example, variations in atmospheric conditions.

Graphing Calculator Usage

Graphing calculators are an invaluable tool for visualizing and solving complex mathematical problems, as they can plot functions, calculate numerical values, and find intersection points. In our hurricane model, after finding the theoretical solution, we can use a graphing calculator to visually verify that our solution is reasonable.

Using features like \([\mathrm{TRACE}]\) and \([\mathrm{ZOOM}]\), or by computing the intersect command, a student can see the exact point on the graph where the barometric air pressure reaches the specified value of 29 inches of mercury. This not only helps with understanding the problem at hand but also solidifies the student's comprehension of the relationship between the mathematical model and its graphical representation.

Graphing calculators can also be used to explore behavior of functions by tweaking parameters, thereby offering a hands-on experience with mathematical concepts. As an educational tool, the ability to immediately visualize the impact of changes to an equation's parameters is immensely powerful and fosters a deeper understanding of the material.