Problem 133 - Algebra and Trigonometry

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 barometric air pressure increases as the distance from the eye of the hurricane increases, but at a decreasing rate.

Step 1: Understand the Given Function

In this step, observe function \(f(x)=0.48 \ln (x+1)+27\). This function represents the barometric air pressure, \(f(x)\), in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function involves a natural logarithm, so the function may be expected to increase at a decreasing rate.

Step 2: Set Up the Viewing Rectangle

A viewing rectangle \([0,500,50]\) by \([27,30,1]\) is specified in the problem. This means on the x-axis, values are from 0 to 500 with increments of 50 (representing the distance from the hurricane eye in miles), and on the y-axis, values are from 27 to 30 with increments of 1, representing the barometric air pressure in inches of mercury.

Step 3: Graph the Function

Using the above viewing rectangle, graph the function \(f(x)=0.48 \ln (x+1)+27\). The graph will be a curve that increases as the distance from the eye of the hurricane increases, but at a decreasing rate.

Step 4: Interpret the Graph

Notice the shape of the graph and what it signifies about the barometric air pressure. The graph reveals that as the distance from the eye of the hurricane increases, the barometric air pressure also increases but at a decreasing rate. This means that the farther away one is from the eye of the hurricane, the less sharp the change in barometric air pressure is, but it still keeps increasing.