Problem 49 - A Graphical Approach to Precalculus with Limits

Question

A ship travels 50 kilometers on a bearing of \(27^{\circ}\) and then travels on a bearing of \(117^{\circ}\) for 140 kilometers. Find the distance \(x\) between the starting point and the ending point.

Step-by-Step Solution

Answer

The distance between the starting and ending points is approximately 148.66 km.

Step 1: Sketch the Path

To understand the ship's journey, we start by sketching the path. Draw the starting point as point A. The first leg of the journey takes the ship 50 km on a bearing of \(27^{\circ}\) to point B. From point B, the ship travels 140 km on a bearing of \(117^{\circ}\) to point C. Identify the path as A to B, then B to C.

Step 2: Understand the Bearings

Bearings are measured clockwise from the north. The bearing of \(27^{\circ}\) indicates the angle \(27^{\circ}\) clockwise from the north. Similarly, the bearing of \(117^{\circ}\) indicates an angle of \(117^{\circ} - 27^{\circ} = 90^{\circ}\) relative to the first path AB.

Step 3: Use the Cosine Rule to Find Distance

Since the angle between the paths is \(90^{\circ}\), we use the Pythagorean theorem to find \(x\), the distance between the starting and ending points. Let the length of \(AB = 50\) km and the length of \(BC = 140\) km. Thus, the equation is:\[ x^2 = 50^2 + 140^2 \]Simplify and solve the equation for \(x\).

Step 4: Calculate Distance x

Now, calculate:\[ x^2 = 50^2 + 140^2 = 2500 + 19600 = 22100 \]Take the square root of both sides to find \(x\):\[ x = \sqrt{22100} \approx 148.66 \]Therefore, the distance between the starting and ending point is approximately 148.66 km.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is mainly used to solve problems involving right-angled triangles, but it can also be extended to non-right-angled triangles using specific rules and formulas. In the context of navigation, understanding bearings and how they relate to directional travel is crucial.

• Bearings are a practical application of trigonometry in navigation, for measuring angles from a fixed point, typically north.
• Angles are measured in degrees, clockwise from the north, allowing navigators and pilots to determine precise directions.

For solving problems involving bearings in this exercise, we interpret the paths the ship took as parts of a larger, often right-angled, triangle. This allows trigonometry to effectively calculate distances between points.

Cosine Rule

While the original exercise primarily utilized the Pythagorean theorem—reserved for right-angled triangles—there's another powerful trigonometric tool called the cosine rule. This rule extends the principles of trigonometry to any triangle, not just those with a right angle. It's especially useful in situations where two sides and the included angle are known, but this wasn't necessary in the current problem as the two directions are perpendicular.

• The Cosine Rule is given by the formula: at this source
• It is best used when dealing with triangles where Pythagorean Theorem doesn't apply.
• It can calculate an unknown side when two sides and the angle between them are known.

While not applicable for right angles, understanding this rule can better equip you to handle complex trigonometric problems.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that explains the relationship between the lengths of sides in right-angled triangles. Formulated as \( a^2 + b^2 = c^2 \), where \( c \) represents the hypotenuse, it provides a way to calculate one side of a triangle if the other two are known. In the context of our exercise, the ship's travel formed a right triangle, enabling straightforward application of this theorem.

• Ideal for scenarios where a right angle is present.
• It simplifies the calculation of distances in navigation, construction, and various fields.
• Always requires knowing at least two sides of the triangle to find the third.

In this specific exercise, knowing the paths were perpendicular (forming a 90-degree angle) allowed us to efficiently find the distance from start to end point without needing additional complex calculations.