Problem 84 - College Algebra

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 this function to solve Exercises \(83-84\) 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 distance from the eye of the hurricane where the barometric air pressure is 29 inches of mercury is approximately \(e^{2/0.48} - 1\) miles.

Step 1: Understand the question and write down the equation

The barometric pressure is given to be 29, we have to find the value of \(x\) (distance from the eye of the hurricane). Thus, we set the function equal to 29, that is, \(0.48 \ln(x+1) + 27 = 29\).

Step 2: Isolate the logarithmic term

Our next task is to isolate the \(\ln(x + 1)\) term. It involves simple algebra: subtract 27 from both sides. \(0.48 \ln(x+1) = 29 - 27 = 2\).

Step 3: Isolate the natural logarithm

Next, we aim to isolate \(\ln(x + 1)\) by dividing both sides by 0.48. So, \(\ln(x+1) = \frac{2}{0.48}\).

Step 4: Apply inverse of natural logarithm to both sides

To cancel out the natural logarithm and find \(x\), take \(e^{\text{value}}\) on both sides of the equation. So, \(x + 1 = e^{2/0.48}\).

Step 5: Isolate x

Now, just subtract 1 from both sides of the equation to isolate \(x\). Thus, \(x = e^{2/0.48} - 1\)

Key Concepts

Understanding Hurricanes

Hurricanes are immense atmospheric phenomena characterized by intense winds and heavy rainfall. These storms form over warm ocean waters and are known for their destructive capabilities. Several factors contribute to their formation and intensity. First, warm temperatures at the water's surface provide the energy required to initiate the storm system. As the warm and moist air rises, it creates a low-pressure center known as the "eye" of the hurricane. This area is often calm, with the most severe weather conditions occurring in the surrounding "eye wall."
Key features of hurricanes include:

• Strong winds exceeding 74 mph (119 km/h).
• Heavy rainfall, leading to flooding.
• Spiral rainbands that extend hundreds of miles from the center.

Understanding these elements helps in predicting and modeling their behavior. Scientists use functions and mathematical models to analyze data, such as barometric pressure, to better predict the impact zones and potential severity of these tropical storms.

Barometric Pressure Role in Hurricanes

Barometric pressure, a measure of atmospheric pressure, plays a crucial role in understanding hurricanes. Within a hurricane, barometric pressure is significantly lower compared to surrounding areas. This low pressure is central to the hurricane's formation and is a vital indicator of its intensity. Typically, the lower the barometric pressure, the stronger the hurricane.
The eye of a hurricane exhibits the lowest pressure, which can be measured in inches of mercury. As one moves away from the eye, the barometric pressure generally increases. Modeling these changes using mathematical functions allows meteorologists to predict how the hurricane will behave. In mathematical terms, as seen in the exercise, a function like \(f(x)=0.48 \ln(x+1)+27\) can help determine the relationship between distance from the eye and pressure.
This relationship is particularly useful for:

• Predicting wind speeds.
• Determining the storm's track.
• Assessing potential impacts on coastal regions.

Understanding this variable aids in forecasting and preparing for such natural disasters.

Using Graphing Utilities

Graphing utilities are essential tools for visualizing complex mathematical relationships, such as those found in hurricane-related models. They allow users to enter functions and explore their behaviors graphically, which benefits both students and meteorologists. In the context of the exercise, a graphing utility can visually display the function \(f(x)=0.48 \ln(x+1)+27\) and determine points of interest, such as when the barometric air pressure is 29 inches of mercury.
Utilizing features like "TRACE" and "ZOOM" can:

• Help users follow the curve's progression and find specific values, such as where the curve meets a particular line.
• Offer a clearer understanding of function behavior over different intervals.
• Allow for the intersection command to find precise solutions.

These capabilities enhance problem-solving skills by providing a visual approach to understanding and verifying solutions, a critical aspect of interpreting real-world data in science and mathematics.