Problem 134 - 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 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 approximately 29 inches of mercury at a distance of about 4.1 miles from the eye of the hurricane.

Step 1: Set function to given value

The barometric air pressure is given as 29 inches of mercury. This is the value of the function \(f(x)\). Therefore, setting the function \(f(x)\) to 29 gives us the equation \(0.48 \ln(x+1)+27 = 29\)

Step 2: Isolate the logarithmic term

Subtract 27 from both sides of the equation: \(0.48 \ln(x+1) = 2\)

Step 3: Remove the coefficient of the logarithmic term

To remove the 0.48 coefficient, divide both sides of the equation by 0.48: \(\ln(x+1) = 2/0.48\)

Step 4: Solve for 'x' in the equation

Apply the anti-logarithmic rule to both sides to isolate \(x+1\): \(x+1 = e^{2/0.48}\)

Step 5: Subtract 1 from both sides

Subtract 1 from each side of the equation to solve for \(x\): \(x = e^{2/0.48} - 1\)

Step 6: Calculation

Next, solve the above equation for \(x\). The answer after calculation should be \(x \approx 4.1\)

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and can be useful in modeling real-world phenomena, such as sound intensity, earthquake strength, and atmospheric pressure. In logarithmic functions, the logarithm of a number is the exponent to which the base must be raised to get that number. For example, in the function \(f(x) = \text{log}_b(x)\), \(b\) is the base. If \(f(x) = c\), then this means \(b^c = x\).

For the hurricane modeling exercise, the logarithmic function \(f(x)=0.48 \text{ln}(x+1)+27\) is provided, where \(\text{ln}\) refers to the natural logarithm, i.e., logarithm to the base \(e\), which is approximately 2.71828. It represents the barometric air pressure in inches of mercury at a given distance \(x\) from the eye of a hurricane. Understanding the behavior and properties of logarithmic functions is crucial for solving the exercise given.

Solving Logarithmic Equations

Solving logarithmic equations typically involves isolating the logarithmic expression and then converting the logarithmic equation to its equivalent exponential form. In the exercise, we have the equation \(0.48 \text{ln}(x+1)+27 = 29\). To solve this, one must isolate the logarithmic term and then use properties of logarithms to find \(x\).

First, subtract 27 from both sides to get \(0.48 \text{ln}(x+1) = 2\), and then divide by 0.48 to isolate the logarithmic term. Afterwards, by applying the anti-logarithm, the equation can be converted to its exponential form: \(x+1 = e^{2/0.48}\). Finally, subtract 1 from both sides to solve for \(x\), giving us the solution. Adequate knowledge of logarithmic properties and operations is essential to navigate through these steps efficiently.

Hurricane Modeling in Algebra

Algebraic models, such as the provided logarithmic function for hurricane air pressure, are tools that allow scientists and mathematicians to represent complex natural phenomena in a simplified mathematical manner. These models enable the prediction of certain behaviors, like the change in air pressure with distance from a hurricane's eye.

Using the given model \(f(x)=0.48 \text{ln}(x+1)+27\), students can calculate the barometric pressure at varying distances from the eye. The specific task to find out how far from the hurricane's eye the air pressure is 29 inches of mercury involves setting \(f(x)\) to 29 and solving for \(x\). Algebraic and logarithmic understanding is critical to interpreting and solving such real-life problems using mathematical models.