This geography problem is solved step by step below, with detailed explanations to help you understand the method and arrive at the correct answer.
ScanSolve AI Answer
Here's how to find the longitude of Q: First, let's define the given information: Point P: (30^ N, 40^ W) Distance PQ: 4500 km Flight direction: Due east, on the same latitude. This means the latitude of Q is also 30^ N. We will use the Earth's radius R = 6371 km. The distance along a parallel of latitude is given by the formula: L = ()/(360^) × 2 R where: L is the distance along the parallel (4500 km) is the difference in longitude in degrees R is the radius of the Earth (6371 km) is the latitude (30^) Step 1: Draw an appropriate diagram. A diagram showing the Earth, the Equator, the 30^ N parallel of latitude, and the positions of P and Q. [scale=0.8] % Earth (0,0) circle (3cm); % Equator (-3,0) -- (3,0) node[right] Equator; % Prime Meridian (simplified) (0,-3) -- (0,3) node[above] PM; % Latitude 30N [dashed] (0, 3sin(30)) ellipse (3cos(30) and 0.5cm); % Ellipse for 30N parallel at (3cos(30), 3sin(30)) [right] 30^ N; % Point P (40W) (P) at (3cos(30)cos(40), 3cos(30)sin(40)); (P) circle (1.5pt) node[below left] P (40^ W); [dashed] (0,0) -- (P); % Meridian line for P % Point Q (East of P) % We'll calculate the angle first, then place Q % For now, just indicate direction at (0, 3*sin(30)) ; % Center of the parallel [->, thick] (P) arc[start angle=180-40, end angle=180-40+46.7, radius=3*cos(30)] node[midway, above] 4500 km; % Calculate Q's position for the diagram _deg46.7 _Q_deg-40 + _deg (Q) at (3cos(30)cos(-_Q_deg), 3cos(30)sin(-_Q_deg)); (Q) circle (1.5pt) node[below right] Q (_Q); [dashed] (0,0) -- (Q); % Meridian line for Q % Angle theta at the pole (simplified representation) [<->] (3cos(30)cos(40)0.8, 3cos(30)sin(40)0.8) arc[start angle=180-40, end angle=180-40+46.7, radius=3cos(30)0.8]; at (3cos(30)cos(40-46.7/2)0.9, 3cos(30)sin(40-46.7/2)0.9) ; Step 2: Substitute the known values into the formula. 4500 = ()/(360^) × 2 (6371 km) (30^) Step 3: Calculate (30^). (30^) = sqrt(3)2 ≈ 0.866025 Step 4: Solve for . 4500 = ()/(360^) × 2 (6371) (0.866025) 4500 = × (2 × 6371 × 0.866025)/(360) 4500 = × (34680.16)/(360) 4500 = × 96.3338 = (4500)/(96.3338) ≈ 46.71^ This is the angular difference in longitude between P and Q. Step 5: Determine the longitude of Q. Point P is at 40^ W. The aeroplane flies due east, so the longitude increases (moves towards 0^ and then into the Eastern hemisphere). Longitude of Q = Longitude of P + (when moving east) Longitude of Q = 40^ W - 46.71^ (treating West as negative and East as positive, or simply moving 46.71^ east from 40^ W) Starting at 40^ W, moving 40^ east brings us to the Prime Meridian (0^). The remaining distance to travel east is 46.71^ - 40^ = 6.71^. Since we crossed the Prime Meridian and continued east, the longitude of Q is 6.71^ E. Step 6: Round to the nearest degree. 6.71^ E ≈ 7^ E. The longitude of Q is 7^ E. That's 2 down. 3 left today — send the next one.