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
You are correct in identifying the ambiguity regarding ship Z's position. Let H be the harbour. Ship X is 16 km from H on a bearing of 090^. So, HX = 16 km. Ship Y is 14 km from H on a bearing of 130^. So, HY = 14 km. Ship Z is described with two pieces of information: "26.31 Km to the west of Y" and "on a bearing of 240^ from the harbour". These two pieces of information are contradictory if interpreted strictly. For typical bearing problems, when a distance and bearing from the harbour are given for a ship, that distance refers to the distance from the harbour. Therefore, we will assume that the distance of Z from the harbour is HZ = 26.31 km, and its bearing from the harbour is 240^. The information "26.31 Km to the west of Y" is considered extraneous or a misstatement, as it creates a contradiction with the given bearing from the harbour. a) The distance between X and Y Step 1: Find the angle XHY. The bearing of X from H is 090^. The bearing of Y from H is 130^. The angle between HX and HY is the difference in their bearings. XHY = 130^ - 090^ = 40^ Step 2: Use the Cosine Rule to find the distance XY. In XHY, we have HX = 16 km, HY = 14 km, and XHY = 40^. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. XY^2 = HX^2 + HY^2 - 2(HX)(HY) ( XHY) XY^2 = 16^2 + 14^2 - 2(16)(14) (40^) XY^2 = 256 + 196 - 448 (40^) XY^2 = 452 - 448(0.7660) XY^2 = 452 - 343.168 XY^2 = 108.832 XY = sqrt(108.832) XY ≈ 10.432 km The distance between X and Y is approximately 10.43 km. b) The distance from the harbour This part of the question asks for "The distance from harbour:". Based on the problem statement and our assumption for Z, this refers to the distance of Z from the harbour. The distance of Z from the harbour H is given as HZ = 26.31 km. The distance from the harbour to Z is 26.31 km. c) The distance between X and Z Step 1: Find the angle XHZ. The bearing of X from H is 090^. The bearing of Z from H is 240^. The angle between HX and HZ is the difference in their bearings. XHZ = 240^ - 090^ = 150^ Step 2: Use the Cosine Rule to find the distance XZ. In XHZ, we have HX = 16 km, HZ = 26.31 km, and XHZ = 150^. XZ^2 = HX^2 + HZ^2 - 2(HX)(HZ) ( XHZ) XZ^2 = 16^2 + (26.31)^2 - 2(16)(26.31) (150^) XZ^2 = 256 + 692.2261 - 841.92 (150^) XZ^2 = 948.2261 - 841.92(-0.8660) XZ^2 = 948.2261 + 729.00992 XZ^2 = 1677.23602 XZ = sqrt(1677.23602) XZ ≈ 40.954 km The distance between X and Z is approximately 40.95 km. 3 done, 2 left today. You're making progress.