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
Okay, let's finish this problem. a) Calculate the distance of Q from P. Step 1: Determine the angle at the base camp R. P moves on a bearing of 205° from R. Q moves on a bearing of 060° from R. The angle PRQ between their paths from R is the difference between these bearings. PRQ = 205^ - 060^ = 145^ Step 2: Apply the Cosine Rule to find the distance PQ. We have a triangle RPQ with sides RP = 20 km, RQ = 15 km, and the included angle PRQ = 145^. PQ^2 = RP^2 + RQ^2 - 2(RP)(RQ)( PRQ) PQ^2 = 20^2 + 15^2 - 2(20)(15)(145^) PQ^2 = 400 + 225 - 600(-0.819152) PQ^2 = 625 + 491.4912 PQ^2 = 1116.4912 PQ = sqrt(1116.4912) ≈ 33.4139 km Rounding to the nearest whole number: PQ ≈ 33 km b) Calculate the bearing of Q from P. Step 3: Use the Sine Rule to find RPQ. To find the bearing of Q from P, we first need to determine an angle within RPQ. Let's find RPQ. (RQ)/(( RPQ)) = (PQ)/(( PRQ)) (15)/(( RPQ)) = (33.4139)/((145^)) ( RPQ) = (15 (145^))/(33.4139) ( RPQ) = (15 × 0.573576)/(33.4139) ( RPQ) = (8.60364)/(33.4139) ≈ 0.25747 RPQ = (0.25747) ≈ 14.92^ Step 4: Calculate the bearing of Q from P. The bearing of P from R is 205°. The back bearing of R from P is 205^ - 180^ = 25^. This is the angle from the North line at P, measured clockwise to the line segment PR. The bearing of Q from P is the angle from the North line at P, measured clockwise to the line segment PQ. Bearing of Q from P = (Back bearing of R from P) + RPQ Bearing of Q from P = 25^ + 14.92^ Bearing of Q from P = 39.92^ Rounding to the nearest whole number and expressing as a three-figure bearing: Bearing of Q from P ≈ 040^ That's 2 down. 3 left today — send the next one.