Problem 43 - Precalculus Mathematics for Calculus

Question

A pilot flies in a straight path for \(1 \mathrm{h} 30 \mathrm{min} .\) She then makes a course correction, heading \(10^{\circ}\) to the right of her original course, and flies \(2 \mathrm{h}\) in the new direction. If she maintains a constant speed of \(625 \mathrm{mi} / \mathrm{h}\), how far is she from her starting position?

Step-by-Step Solution

Answer

The pilot is approximately 367 miles from her starting position.

Step 1: Calculate the Distance for Each Flight Segment

First, let's calculate the distance the pilot flies during each segment of her flight. For the first segment, she flies for \(1.5\) hours at a speed of \(625 \text{ mi/h}\). The distance is given by the formula \(\text{Distance} = \text{Speed} \times \text{Time}\). Thus, the first segment distance is \(1.5 \times 625 = 937.5 \text{ miles}\).For the second segment, she flies for \(2\) hours at the same speed. Therefore, the distance is \(2 \times 625 = 1250 \text{ miles}\).

Step 2: Use the Law of Cosines to Find the Total Distance

The pilot’s path forms a triangle where the two sides are the distances of the segments flown: \(937.5\) miles and \(1250\) miles, with an included angle of \(10^{\circ}\).The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab\cos(C),\]where \(c\) is the distance from the starting position, \(a = 937.5\) miles, \(b = 1250\) miles, and \(C = 10^{\circ}\).Substitute into the formula:\[ c^2 = 937.5^2 + 1250^2 - 2 \times 937.5 \times 1250 \times \cos(10^{\circ})\]Calculate this to find \(c^2\).

Step 3: Calculation and Result

Substituting the given values and computing gives:\[ c^2 = 878906.25 + 1562500 - 2343750 \cdot \cos(10^{\circ}).\]Using a calculator, \(\cos(10^{\circ}) \approx 0.9848\), so:\[ c^2 = 878906.25 + 1562500 - 2343750 \times 0.9848 \]\[ c^2 = 878906.25 + 1562500 - 2307503.75 \]\[ c^2 = 134902.5\]\[ c \approx \sqrt{134902.5} \approx 367 \text{ miles}\]Therefore, the distance from her starting position is approximately \(367\) miles.

Key Concepts

Understanding the Law of Cosines

The Law of Cosines is a crucial theorem in trigonometry, and it helps to relate the lengths of the sides of a triangle to the cosine of one of its angles. This is particularly helpful in calculating distances when we know some angles and side lengths, which is often the case in navigation and surveying.

Essentially, the Law of Cosines gives us the formula:

• \[ c^2 = a^2 + b^2 - 2ab\cos(C) \]

In this formula:

• \( c \) is the length of the side opposite the angle \( C \)
• \( a \) and \( b \) are the lengths of the other two sides
  
• \( C \) is the angle between sides \( a \) and \( b \)

This rule becomes especially useful when dealing with non-right angled triangles where we cannot use simpler trigonometric relationships like the Pythagorean theorem. For instance, in the exercise we're discussing, the pilot's path formed a triangle with sides 937.5 miles and 1250 miles, and a 10-degree angle between these paths. Using the Law of Cosines, the distance from the starting point was calculated to be approximately 367 miles.

The Law of Cosines is not just a bunch of numbers; it's a bridge that connects geometry and trigonometry, enabling us to solve real-world problems involving triangular paths and structures.

Mastering Distance Calculation

Calculating distance is a fundamental skill in mathematics, especially in navigation and physics. In our case, it involves using the basic formula for distance, which is

• \( \text{Distance} = \text{Speed} \times \text{Time} \)
  

This formula is straightforward yet powerful. In the exercise, we used it to find how far the pilot flew during each part of her journey:

• For the initial flight of 1.5 hours at 625 miles per hour, the distance was 937.5 miles (1.5 hours \(\times\) 625 mph).
  
• For the second segment of 2 hours at the same speed, the distance was 1250 miles (2 hours \(\times\) 625 mph).

Once you have these individual distances, they serve as the sides of a triangle (as in our example). These calculations are the stepping stones to using more complex trigonometric formulas like the Law of Cosines. This approach is not isolated to this problem alone; understanding how to break down and calculate each segment's distance helps solve a wide array of problems, whether it's in navigation related applications or any task involving moving objects at a steady speed over time.

Basics of Angle Measurement

Understanding angle measurement is essential, especially when navigating or manipulating geometric figures. Angles are measured in degrees, with a full circle representing 360 degrees. For our pilot example, a course correction involved a change of 10 degrees to the right.

It’s important to grasp that the direction in which an angle is measured can determine the triangle's configuration and thus influence subsequent calculations. In the problem, the angle change was integral to redefining the direction of the pilot's flight and formed the basis for applying the Law of Cosines to measure the overall triangular route.

Measuring angles accurately often involves various tools such as protractors and digital devices. However, working with them mathematically often requires an understanding of trigonometric functions such as sine, cosine, and tangent. These functions are vital as they help describe relationships within triangles. For instance, the cosine function was used here to solve the triangle and find the length of the side opposite to our designated angle.

Developing a strong foundational understanding of angle measurement paves the way for more advanced applications in navigation, architecture, and even in fields such as astronomy, where precision in angle measurement is paramount.