This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve this problem, we'll use trigonometry, specifically the Cosine Rule and Sine Rule, after determining the angles in the triangle formed by towns A, B, and C. Part (a): Shortest distance from A to C Step 1: Draw a diagram and identify the knowns. Let town A be the origin. A man travels 5 km due east from A to B. So, the distance AB = 5 km. From B, he travels 7 km to C on a bearing of 162^. So, the distance BC = 7 km. We need to find the angle ABC. Draw a North line at B. Since A is due west of B (because B is due east of A), the line BA points in the direction of 270^ from the North line at B. The bearing of C from B is 162^. The angle ABC is the angle between the line BA and the line BC. ABC = 270^ - 162^ = 108^ Step 2: Apply the Cosine Rule to find the length AC. The Cosine Rule states AC^2 = AB^2 + BC^2 - 2(AB)(BC) ( ABC). AC^2 = 5^2 + 7^2 - 2(5)(7) (108^) AC^2 = 25 + 49 - 70 (108^) AC^2 = 74 - 70(-0.30901699) AC^2 = 74 + 21.6311893 AC^2 = 95.6311893 AC = sqrt(95.6311893) AC ≈ 9.7791209 km Step 3: Round the distance to the nearest km. AC ≈ 10 km Part (b): Bearing of C from A Step 4: Use the Sine Rule to find BAC. The Sine Rule states ( A)/(a) = ( B)/(b). (( BAC))/(BC) = (( ABC))/(AC) (( BAC))/(7) = ((108^))/(9.7791209) ( BAC) = (7 (108^))/(9.7791209) ( BAC) = (7 × 0.9510565)/(9.7791209) ( BAC) ≈ 0.68081 BAC = (0.68081) BAC ≈ 42.90^ Step 5: Calculate the bearing of C from A. Town B is due east of Town A. This means the line AB is along the East direction from A. The angle from the North line at A to the East direction (line AB) is 90^. The bearing of C from A is the angle measured clockwise from the North line at A to the line AC. Bearing of C from A = 90^ + BAC. Bearing = 90^ + 42.90^ = 132.90^ Step 6: Round the bearing to the nearest degree. Bearing ≈ 133^ What's next?