Problem 85 - Algebra and Trigonometry

Question

The wind is blowing at 10 knots. Sailboat racers look for a sailing angle to the 10-knot wind that produces maximum sailing speed. In this application, (r, \(\theta\)) describes the sailing speed, r, in knots, at an angle \(\theta\) to the 10-knot wind. Use this information to solve. Interpret the polar coordinates: \(\left(6.3,50^{\circ}\right)\)

Step-by-Step Solution

Answer

The sailboat is moving with a speed of 6.3 knots at an angle of 50 degrees to the 10-knot wind.

Step 1: Understand Polar Coordinates

In Polar Coordinates, a point is represented by the radius and the angle. In this context, the radius represents the speed of the sailboat in knots, while the angle represents the direction relative to the 10-knot wind. This is what the given polar coordinates \(\left(6.3,50^\circ\right)\) mean in this situation.

Step 2: Interpret the given coordinates

The given polar coordinates are \(\left(6.3,50^\circ\right)\). This means that the sailboat is moving at a speed of 6.3 knots. The angle of 50° is the direction the sailboat is making relative to the 10-knot wind. Specifically, it is 50° away from the direction of the wind.

Key Concepts

Sailing Speed

Understanding sailing speed is essential for sailboat racers who are constantly seeking to optimize their performance. The idea of sailing speed in polar coordinates involves the radius (\( r \)), which provides the boat's speed relative to the water.

In this context, the sailboat's speed is given in knots, a nautical unit for measuring speed. The faster a sailboat can travel in the given wind conditions, the more competitive it can be in a race.

• The radius indicates the scalar quantity or sailing speed.
• Here, a radius of 6.3 signifies a sailboat speed of 6.3 knots.

The speed is crucial because it determines how efficiently a boat can cover distances and maneuver in relation to the direction and strength of the wind.

Sailing Angle

The sailing angle in the context of polar coordinates is just as critical as speed for a sailboat's navigation and performance.

The angle (\( \theta \)) describes the direction that the boat is traveling relative to the wind direction. A good sailing angle can dramatically impact a sailboat's speed and trajectory.

• An angle of \( 50^{\circ} \) means that the boat's path is at a 50-degree angle to the direction of the 10-knot wind.
• Strategically choosing an angle can help capture the best wind to maximize speed.

Understanding and refining the correct angle allows sailors to harness the wind efficiently and sustain optimal speeds during a race.

Wind Speed

Wind speed serves as a backdrop for determining sailing speed and angle. In this exercise, the wind is constant at 10 knots.

Sailboat performance and choices hinge significantly on this factor because the wind provides the power needed for movement.

• The higher the wind speed, the faster the potential sailing speed.
• However, it is essential to balance sail settings and sailing angle for maximum efficiency.

The interplay between sailing angles and the constant 10-knot wind proves crucial in finding the optimal path and harnessing nature's resources to competitively lead a race.

Radius and Angle Interpretation

In terms of polar coordinates, interpreting the radius and angle helps visualize the boat’s sailing conditions as a point or position in a specific coordinate system. This not only provides clarity for performance expectations but also directs precise navigation.

When viewing \((6.3, 50^{\circ})\), the coordinates mean:

• 6.3 knots is the sailboat's speed – represented as the radius.
• 50 degrees is the direction relative to the wind – represented as the angle.

Through these interpretations, sailors can strategically plan how to progress their races by adjusting angles and speeds for the best outcomes relative to the wind’s behavior.