Question

trigonometry and directional bearing
solve each problem by filling in the figure with all required information, performing the necessary calculations, and writing your final answer in a complete sentence with appropriate units.

1. an airplane is flying from city a to city b at a bearing of 100°. the distance between the two cities is 1,200 miles. how far west is city a relative to city b? round the answer to the nearest mile.

use the following graph to set up and solve the problem. let the origin be city a. the perpendicular symbol around the origin is assumed and has been removed.
graph: n (up), s (down), w (left), e (right), origin at center labeled a
a) set up the problem by adding the given information to the graph.
given information: the bearing and the distance between two cities
b) fill in the rest of the needed information onto the graph.
needed information: the unknown angle measure and unknown distance

Explanation:

Step1: Determine the angle with x - axis

Bearing of \(100^{\circ}\) means the angle from the north direction towards the east is \(100^{\circ}\). So the angle between the line \(AB\) (where \(A\) is origin and \(B\) is the position of city \(B\)) and the negative \(x\) - axis (west - east axis, west is negative \(x\)) is \(100^{\circ}-90^{\circ} = 10^{\circ}\). Let the distance of city \(A\) west of city \(B\) be \(x\). We can use the cosine function in the right - triangle formed (where the hypotenuse is the distance between \(A\) and \(B\) (\(c = 1200\) miles), and the adjacent side to the angle of \(10^{\circ}\) is \(x\)). The cosine of an angle \(\theta\) in a right - triangle is given by \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 10^{\circ}\), adjacent side is \(x\), and hypotenuse \(c = 1200\).

Step2: Calculate the value of \(x\)

We know that \(\cos(10^{\circ})=\frac{x}{1200}\). So, we can solve for \(x\) by multiplying both sides of the equation by \(1200\): \(x = 1200\times\cos(10^{\circ})\).
We know that \(\cos(10^{\circ})\approx0.9848\). Then \(x = 1200\times0.9848 = 1181.76\approx1182\) miles.

Answer:

City A is approximately 1182 miles west of City B.