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
a) Step 1: Determine the coordinates of town T from John's initial position. Let John's initial position be the origin O(0,0). John travels 40 km on a bearing of 210^. To convert a bearing to a standard angle (measured counter-clockwise from the positive x-axis), we use = 90^ - bearing. For a bearing of 210^: _T = 90^ - 210^ = -120^ The coordinates of T are (x_T, y_T): x_T = 40 (-120^) = 40 (-(1)/(2)) = -20 y_T = 40 (-120^) = 40 (-sqrt(3)2) = -20sqrt(3) So, the coordinates of town T are (-20, -20sqrt(3)). Step 2: Determine the displacement of Attah from town T. Attah moves 35 km from T on a bearing of 300^. For a bearing of 300^: _A = 90^ - 300^ = -210^ The displacement vector from T to Attah's final position is (x_disp, y_disp): x_disp = 35 (-210^) = 35 (150^) = 35 (-sqrt(3)2) = -17.5sqrt(3) y_disp = 35 (-210^) = 35 (150^) = 35 ((1)/(2)) = 17.5 Step 3: Calculate Attah's final position from John's initial position. Attah's final position (x_Attah, y_Attah) is the sum of T's coordinates and Attah's displacement from T: x_Attah = x_T + x_disp = -20 - 17.5sqrt(3) y_Attah = y_T + y_disp = -20sqrt(3) + 17.5 Using sqrt(3) ≈ 1.732: x_Attah = -20 - 17.5(1.732) = -20 - 30.31 = -50.31 y_Attah = -20(1.732) + 17.5 = -34.64 + 17.5 = -17.14 The position of Attah from John's initial position is approximately (-50.31, -17.14) km. The position of Attah from John's initial position is (-50.31, -17.14). b) Given A(1,2) and the vectors AB = 3 \\ 2 , BC = -1 \\ 3 , and CD = 2 \\ -2 . Step 1: Calculate the coordinates of B. The position vector of B is OB = OA + AB. OB = 1 \\ 2 + 3 \\ 2 = 1+3 \\ 2+2 = 4 \\ 4 So, the coordinates of B are (4,4). Step 2: Calculate the coordinates of C. The position vector of C is OC = OB + BC. OC = 4 \\ 4 + -1 \\ 3 = 4-1 \\ 4+3 = 3 \\ 7 So, the coordinates of C are (3,7). Step 3: Calculate the coordinates of D. The position vector of D is OD = OC + CD. OD = 3 \\ 7 + 2 \\ -2 = 3+2 \\ 7-2 = 5 \\ 5 So, the coordinates of D are (5,5). 3 done, 2 left today. You're making progress.