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 will use the Cosine Rule and Sine Rule, along with principles of bearings. First, let's define the given information: • Distance PQ = 50 km • Distance PR = 90 km • Bearing of Q from P = 075^ • Bearing of R from P = 310^ Step 1: Determine the interior angle QPR. The bearing of Q from P is 075^. The bearing of R from P is 310^. The angle between the lines PQ and PR, measured clockwise from PQ to PR, is 310^ - 075^ = 235^. This is the reflex angle. The interior angle QPR is 360^ - 235^ = 125^. Step 2: Calculate the distance QR using the Cosine Rule (Part a). In PQR, we have two sides (PQ and PR) and the included angle ( QPR). We can use the Cosine Rule: QR^2 = PQ^2 + PR^2 - 2(PQ)(PR) ( QPR) Substitute the known values: QR^2 = 50^2 + 90^2 - 2(50)(90) (125^) QR^2 = 2500 + 8100 - 9000 (125^) We know that (125^) = -(180^ - 125^) = -(55^). (55^) ≈ 0.573576 QR^2 = 10600 - 9000(-0.573576) QR^2 = 10600 + 5162.184 QR^2 = 15762.184 QR = sqrt(15762.184) QR ≈ 125.5475 km Rounding to two decimal places: The distance between Q and R is 125.55 km. Step 3: Calculate PQR using the Sine Rule. Now we need to find an angle within PQR to help determine the bearing. Let's find PQR. Using the Sine Rule: (( PQR))/(PR) = (( QPR))/(QR) ( PQR) = (PR ( QPR))/(QR) ( PQR) = (90 (125^))/(125.5475) We know (125^) = (180^ - 125^) = (55^). (55^) ≈ 0.819152 ( PQR) = (90 × 0.819152)/(125.5475) ( PQR) = (73.72368)/(125.5475) ( PQR) ≈ 0.58729 PQR = (0.58729) PQR ≈ 35.96^ Step 4: Calculate the bearing of R from Q (Part b). First, find the back bearing of P from Q. The bearing of Q from P is 075^. The back bearing of P from Q is 075^ + 180^ = 255^. This means the line segment QP points in the direction 255^ from the North line at Q. Now, we need to determine if PQR is added to or subtracted from the bearing of P from Q. From P, Q is at 075^ and R is at 310^. This means R is to the "right" (clockwise) of the line PQ when viewed from P (since 310^ > 075^). Therefore, when standing at Q and looking towards P (bearing 255^), R will be to your "right" (clockwise) relative to the line QP. So, the bearing of R from Q will be the bearing of P from Q plus PQR. Bearing of R from Q = 255^ + PQR Bearing of R from Q = 255^ + 35.96^ Bearing of R from Q = 290.96^ Rounding to the nearest degree (or two decimal places as calculated): The bearing of R from Q is 290.96^. That's 2 down. 3 left today — send the next one.