Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Two boats leave a dock together. Each travels in a straight line. The angle between their courses measures \(54^{\circ} 10^{\prime} .\) One boat travels 36.2 kilometers per hour and the other 45.6 kilometers per hour. How far apart will they be after 3 hours?

Step-by-Step Solution

Answer

After 3 hours, the boats will be approximately 115.15 km apart.

Step 1: Determine the Distance Traveled by Each Boat

To find how far apart the boats will be after 3 hours, start by calculating the distance each boat travels. For the first boat traveling at 36.2 km/h, the distance is \(36.2 \times 3 = 108.6\) kilometers. For the second boat traveling at 45.6 km/h, the distance is \(45.6 \times 3 = 136.8\) kilometers.

Step 2: Understand the Geometric Representation

The two boats create a triangle where the two distances traveled form the two sides, and the angle between them is \(54^{\circ} 10^{\prime}\). This forms a triangle with sides 108.6 km and 136.8 km, and an included angle of \(54^{\circ} 10^{\prime}\).

Step 3: Calculate the Distance Between the Two Boats

Use the law of cosines to find the distance \(d\) between the boats after 3 hours: \[d^2 = 108.6^2 + 136.8^2 - 2 \times 108.6 \times 136.8 \times \cos(54^{\circ} 10^{\prime})\]First, convert the angle from degrees and minutes to just degrees: \(54^{\circ} 10^{\prime} = 54.1667^{\circ}\). Calculate the cosine of this angle and apply it in the formula.

Step 4: Solve for the Distance

Calculate each part of the law of cosines:1. \(108.6^2 = 11792.76\)2. \(136.8^2 = 18706.24\)3. \(2 \times 108.6 \times 136.8 \times \cos(54.1667^{\circ})\)Find \(\cos(54.1667^{\circ})\) using a calculator, which is approximately 0.5852. Now complete the calculation:\[d^2 = 11792.76 + 18706.24 - 2 \times 108.6 \times 136.8 \times 0.5852\]\[d^2 = 11792.76 + 18706.24 - 17245.201\]\[d^2 = 13253.799\]Finally, take the square root: \[d = \sqrt{13253.799} \approx 115.15\] kilometers.

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that helps us understand the relationships between the angles and sides of triangles. It's especially handy when dealing with real-world scenarios, like the problem of our two boats.
When you have a triangle and you know two sides and the angle between them, you can use the law of cosines. This law helps us find unknown lengths. It's particularly useful for non-right-angled triangles.

• The law of cosines formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] where \(c\) is the side opposite angle \(C\), and \(a\) and \(b\) are the other two sides.
• This extension of the Pythagorean theorem works no matter what type of triangle you're working with.
• In our problem, this rule helped calculate the distance between the two boats by considering their path as sides of a triangle and the angle between them.

By utilizing trigonometry fundamentals, you can solve various problems involving distances and angles safely and accurately.

Distance Calculation

Distance calculations aren't just for navigators or mathematicians. They're essential for anyone trying to measure the space between objects. In the boat exercise, you calculated how far each boat travelled over a period, which is vital for the solution.
The math is simple: Multiply speed by time, using the equation:

• Distance = Speed \(\times\) Time
• For example, if Boat A travels at 36.2 km/h, over 3 hours, it travels \(36.2 \times 3 = 108.6\ km\)
• Boat B, at 45.6 km/h, travels \(45.6 \times 3 = 136.8\ km\)

By knowing how far each boat moved, one could form a geometric shape that helps compute their distance apart later on.
These steps might seem basic, but they're a critical component of solving much more complex situations.

Angles in Geometry

Angles play a crucial role in geometry, shaping everything from the simplest shapes to complex structures. In our problem involving two boats, we examined the angle formed between their paths, which was essential to calculate the distance they were apart.
An angle is simply the space between two intersecting lines or surfaces at or close to the point where they meet, measured in degrees.

• A full circle is divided into 360 degrees.
• Sometimes, angles are given in degrees and minutes, as in the boat problem where the angle was \(54^{\circ} 10^{'}\).
• To simplify, this can be converted into decimal degrees: \(54.1667^{\circ}\).

Understanding how to work with angles is vital for solving geometry problems, like the one involving our boats. These angles allowed us to apply the law of cosines accurately, resolving how far apart the boats were after 3 hours of travel. By mastering angles, you can better approach various mathematical and real-world challenges.