Problem 67 - A Graphical Approach to College Algebra

Question

Barometric Pressure The function $$f(x)=27+1.105 \log (x+1)$$ approximates the barometric pressure in inches of mercury at a distance of x miles from the eye of a hurricane. (Source: Miller, A. and R. Anthes, Meteorology, Fifth Edition, Charles E. Merrill.) (a) Approximate the pressure 9 miles from the eye of the hurricane. (b) The ordered pair (99,29.21) belongs to this function. What information does it convey?

Step-by-Step Solution

Answer

(a) The pressure at 9 miles is approximately 28.105 inches. (b) At 99 miles, pressure is 29.21 inches.

Step 1: Understand the Function

The given function is \( f(x) = 27 + 1.105 \log (x+1) \). This represents the barometric pressure, measured in inches of mercury, at a distance \( x \) miles from the eye of a hurricane.

Step 2: Substitute x = 9 into the Function

To approximate the pressure 9 miles from the eye, we substitute \( x = 9 \) in the function: \( f(9) = 27 + 1.105 \log (9+1) \).

Step 3: Simplify the Logarithmic Expression

Calculate \( \log(9+1) = \log(10) \). Since \( \log(10) = 1 \) (using base 10 logarithm), we have \( f(9) = 27 + 1.105 \times 1 \).

Step 4: Calculate the Pressure

Substitute and simplify to find \( f(9) = 27 + 1.105 = 28.105 \). So, the pressure 9 miles away is approximately 28.105 inches of mercury.

Step 5: Interpret the Ordered Pair (99, 29.21)

The ordered pair \((99, 29.21)\) means that at 99 miles from the hurricane's eye, the barometric pressure is 29.21 inches of mercury. This verifies that the function accounts for distances up to 99 miles.

Key Concepts

Logarithmic Functions

Logarithmic functions, commonly referred to as logs, are the inverse of exponential functions. When you see the word 'logarithm,' think "What power must I raise a base number to in order to get another number?"
For example, the expression \( \log_{10}(100) \) answers the question, "To what power must 10 be raised to result in 100?" The answer is 2, because \( 10^2 = 100 \). In other words, \( \log_{10}(100) = 2 \).

Your problem uses the logarithm \( \log(x + 1) \), where we assume a base of 10, which is common in scientific applications.

• Logarithms help to transform multiplicative relationships into additive ones, making calculations more manageable.
• They are particularly useful in engineering, physics, and astronomy for dealing with exponential growth or decay.
• Logarithmic functions graph as a curve that rises slowly and never touches the x-axis, showing how it can be used to model phenomena like sound intensity, earthquakes, and in this case, atmospheric pressure.

Understanding logarithmic properties, such as \( \log(10) = 1 \) because 10 is raised to the power of 1 to get 10, is key in interpreting problems involving tricky log instances.

Hurricane Pressure

Hurricanes are intense tropical weather systems with exceedingly low-pressure centers, also known as the "eye". The pressure in a hurricane can be drastically lower than the surrounding atmospheric pressure, and this variance is what fuels the storm's intensity.

Barometric pressure, measured in inches of mercury, is key to understanding hurricane dynamics. On a clear, calm day at sea level, the normal atmospheric pressure is about 29.92 inches of mercury. During a hurricane, the pressure can fall significantly. A lower pressure indicates a stronger storm; thus, understanding pressure variation across distances from the hurricane's eye is vital.

• The given function, \(f(x) = 27 + 1.105 \log(x+1)\), models how this pressure changes at different distances.
• Closer to the eye, the pressure drop is sharper. As you move away, the pressure gradually increases to normal atmospheric levels.
• The ordered pair (99, 29.21) in your problem signifies that at 99 miles away, the pressure is closer to standard sea level conditions.

It is essential to understand these pressure patterns to accurately predict the storm's potential impact on different regions.

Mathematical Approximation

Mathematical approximation is a practical method used to find close estimates when exact values are difficult or impossible to determine.

In the context of weather and dynamics like hurricanes, exact values for every single point can be complex and computationally heavy. Therefore, mathematical models like the one given in your exercise, \(f(x) = 27 + 1.105 \log(x+1)\), are used to approximate values.

• These models convert complicated real-world phenomena into simpler mathematical expressions that are easier to work with, even if they may not capture every detail.
• Approximation recognizes that small errors in measurement or calculation are acceptable for the sake of simplicity and efficiency.
• Understanding that \(f(9) \approx 28.105\) inches of mercury at 9 miles from a hurricane's eye is useful for quickly assessing areas without computing exact pressure values for each point.

When using approximations, always consider the potential for small inaccuracies but rely on them for a broad understanding of data trends and behaviors.