Problem 131 - Introductory and Intermediate Algebra for College Students 4th

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. Graph the function in a \([0,500,50]\) by \([27,30,1]\) viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the eye increases?

Step-by-Step Solution

Answer

The graph of the function \(f(x)=0.48 \ln(x+1)+27\) increases as the distance from the eye of the hurricane increases. This indicates that the barometric pressure is lower at the hurricane's eye and increases as we move away from it. The rate of increase slows as the distance from the eye increases.

Step 1: Graphing the function

Use graphing software or graphing calculator to sketch the function \(f(x)=0.48 \ln(x+1)+27\). Set the dimensions of the viewing rectangle to [0,500,50] by [27,30,1]. This choice of dimensions allows you to have a detailed view of how the barometric pressure varies with the distance from the hurricane's eye.

Step 2: Analyzing the graph

Take a close look at the graph. The vertical axis (y-axis) measures the barometric pressure while the horizontal axis (x-axis) represents the distance from the eye of the hurricane.

Step 3: Interpretation of the graph shape

From the shape of the graph, it can be observed that, as the distance from the eye of the hurricane increases (i.e., as \(x\) becomes larger), the barometric air pressure \(f(x)\) increases, although not linearly. This indicates that the air pressure is lower at the eye of the hurricane and increases as you move away from the eye. The rate at which this pressure is increasing tends to slow down as we move further away from the eye.