Question

Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is N29.6 \({ }^{\circ} \mathrm{E}\), and the bearing from Oklahoma City to St. Louis is N57.7 \(^{\circ}\) E. The bearing from St. Louis to Oklahoma City is \(S 57.7^{\circ}\) W, and the bearing from St. Louis to Kansas City is \(\mathrm{N} 79.4^{\circ} \mathrm{W}\). How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?

Short answer

The extra frequent flyer miles Adam will receive is 617.3 miles.

Step-by-step solution

- Draw a diagram

Sketch the points for Oklahoma City (OKC), Kansas City (KC), and St. Louis (STL). Draw the bearings from OKC to KC (N29.6°E) and OKC to STL (N57.7°E).

- Determine the bearings

Note that from OKC to STL the bearing is N57.7°E and from OKC to KC it is N29.6°E. Use these bearings to form angle \( \theta \) between the routes.

- Calculate angle \( \theta \)

The angle \( \theta \) between the routes from OKC to STL and OKC to KC is \( 57.7^{\circ} - 29.6^{\circ} = 28.1^{\circ} \).

- Use trigonometry to calculate OKC to KC

Assume the distance from OKC to KC is \( x \) miles. Using Law of Sines in the triangle OKC-STL-KC, apply \[ \frac{x}{\text{sin}(57.7^{\circ})} = \frac{461.1}{\text{sin}(28.1^{\circ})} \].

- Solve for \( x \)

Rearrange the equation to solve for \( x \): \[ x = 461.1 \times \frac{\text{sin}(57.7^{\circ})}{\text{sin}(28.1^{\circ})} \]. Calculate \( x = 461.1 \times \frac{0.848}{0.472} \approx 829.4 \) miles.

- Calculate total distance for connecting flight

Add the distances OKC to KC and KC to STL. \[ 829.4 \text{ miles from OKC to KC} + 249 \text { miles from KC to STL} = 1078.4 \text{ miles} \].

- Determine the extra miles

Determine the extra mileage Adam will collect with the connecting flight as: \[ 1078.4 \text{ total connecting flight miles} - 461.1 \text{ direct flight miles} = 617.3 \text{ extra miles} \].

Key concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It is essential to understand trigonometry for solving problems like the one involving Adam's flight.

In this problem, we use trigonometry to determine unknown distances and angles. The primary function used here is the sine function, which relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.

When dealing with non-right triangles, as seen in Adam’s flight path, trigonometry helps us by providing laws like the Law of Sines to relate the sides and angles of triangles. Understanding basic trigonometric functions - sine, cosine, and tangent - is extremely helpful in solving such exercises.

Law of Sines

The Law of Sines is a crucial tool in trigonometry when dealing with non-right triangles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. It can be written as:

\( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

In the problem involving Adam, we form a triangle with points OKC, STL, and KC. We know the distance from OKC to STL and the angles at these points.

By applying the Law of Sines, we can solve for the unknown distance from OKC to KC. This is done by setting up the equation with known distances and angles, thus allowing us to isolate and calculate the unknown distance.

Distance Calculation

Distance calculation in this context involves using trigonometric laws to work out the lengths between various points on Adam's flight path. The direct distance from OKC to STL is given as 461.1 miles.

By understanding the angles involved and using the Law of Sines, we calculate the additional distance from OKC to KC. We determine the bearing angles and use trigonometric functions to find the combined distance.

This gives us the total miles Adam will travel if he takes the connecting flight, which is essential for determining how many extra frequent flyer miles he will earn compared to the direct flight.

Angle Measurement

Angle measurement is a fundamental concept in solving this type of problem. Understanding how to measure and work with angles helps us in applying trigonometric laws.

In this exercise, the bearings provided allow us to determine the angle between flight routes. Bearings like N29.6°E and N57.7°E help us understand how the flight paths are oriented.

We calculate the angle θ by subtracting the bearings, leading to precise angle measurements necessary for applying the Law of Sines. This step is critical as it simplifies the triangle problem, making it solvable using trigonometric ratios.