Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 62 mph needs 1¼ hr to go from Atlanta to Macon and 1¾ hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

Verified step by step guidance

1. Identify the distances traveled from Atlanta to Macon and from Macon to Augusta using the formula \(\text{distance} = \text{speed} \times \text{time}\). For Atlanta to Macon, calculate \(d_1 = 62 \times 1.25\), and for Macon to Augusta, calculate \(d_2 = 62 \times 1.75\).
2. Draw a diagram representing the positions of Atlanta, Macon, and Augusta, including the bearings. The bearing from Atlanta to Macon is \(S 27^\circ E\), which means starting from south, rotate 27 degrees towards east. The bearing from Macon to Augusta is \(N 63^\circ E\), starting from north, rotate 63 degrees towards east.
3. Determine the angle between the two paths (Atlanta to Macon and Macon to Augusta) at point Macon. Since the bearings are given relative to the cardinal directions, find the interior angle between the two directions by adding or subtracting the given angles appropriately.
4. Use the Law of Cosines to find the distance from Atlanta to Augusta. Label the triangle with sides \(d_1\), \(d_2\), and the unknown side \(d\), and the included angle \(\theta\) found in the previous step. The Law of Cosines formula is: \(d^2 = d_1^2 + d_2^2 - 2 d_1 d_2 \cos(\theta)\).
5. Solve for \(d\) by taking the square root of both sides: \(d = \sqrt{d_1^2 + d_2^2 - 2 d_1 d_2 \cos(\theta)}\). This will give the distance from Atlanta to Augusta.

Key Concepts

Bearing and Direction in Navigation

Bearing is a way to describe direction using angles relative to the cardinal points (N, S, E, W). For example, S 27° E means starting from south and rotating 27° towards east. Understanding how to interpret and convert these bearings into angles for calculations is essential for solving navigation problems.

Distance, Speed, and Time Relationship

The relationship between distance, speed, and time is given by the formula distance = speed × time. Knowing the speed and travel time allows calculation of the distance between two points, which is crucial for determining the lengths of segments in the problem.

Law of Cosines for Non-Right Triangles

The Law of Cosines relates the lengths of sides of any triangle to the cosine of one of its angles. It is used to find an unknown side when two sides and the included angle are known, which is necessary here to find the distance between Atlanta and Augusta given the bearings and distances.