Problem 96 - Algebra and Trigonometry

Question

Suppose that \((r, \theta)\) describes the sailing speed, \(r,\) in knots, at an angle \(\theta\) to a wind blowing at 20 knots. You have a list of all ordered pairs \((r, \theta)\) for integral angles from \(\theta=0^{\circ}\) to \(\theta=180^{\circ} .\) Describe a way to present this information so that a serious sailboat racer can visualize sailing speeds at different sailing angles to the wind.

Step-by-Step Solution

Answer

The provided data can be visualized using a polar plot graph, with the sailing speeds represented by the radial distance from the center of the plot and sailing angles represented by the angle from the initial line. This representation will allow a sailboat racer to interpret the changes in speeds at different sailing angles effectively.

Step 1 - Identifying the Data

From the problem, the data to be observed includes the speed of the sailboat (given in knots) at different angles to the wind (varying from 0 to 180 degrees). The values of r and \(\theta\) form pairs of data points.

Step 2 - Choosing the Appropriate Graph Type

Since the goal is to observe how the sailing speed changes with the angle to the wind, it would be useful to represent this correlation in a form of a polar plot. A polar plot can provide effective visualization of data in terms of radius and angle.

Step 3 - Plotting the Data

On the polar plot, the sailing speeds (r) will be represented by the radial distance from the center, and the angle of sailing (\(\theta\)) will be the angle from the initial line. Each point on the polar plot represents a pair (r, \(\theta\)).

Step 4 - Interpreting the Graph

A serious sailboat racer can easily see the sailing speeds at different angles to the wind by locating the angle on the polar plot and looking at the distance from the center. This way, they can compare which angles provide higher and lower speeds and thus, enhance their racing strategy.