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Pirate Treasure Bearing and Distance Math Problem (Grade 10)
Mathematics
Grade 10 (Junior High School)
Question Content
Some pirates have buried their treasure on an island. The treasure, X, is buried on a bearing of 050° from the swamp, S, and on a bearing of 300° from the caves, C. The caves are 680 m from the swamp on a bearing of 085°. The distance CX is ______ m. The harbour is 650 m south-west of the caves. To find the treasure, the pirates need to walk a distance of ______ m from the harbour on a bearing of ______°. (Give all answers correct to the nearest whole number)
Correct Answer
CX = 523 m; Distance from harbour = 913 m; Bearing = 063°
Detailed Solution Steps
1
Step 1: Calculate angles in triangle SCX. First, convert bearings to internal angles: At point S, the angle between SC (bearing 085°) and SX (bearing 050°) is 85° - 50° = 35°. At point C, the bearing from C to S is 085° + 180° = 265°, so the angle between CS and CX (bearing 300°) is 300° - 265° = 35°. Thus, triangle SCX has angles ∠S=35°, ∠C=35°, so ∠X=110°.
2
Step 2: Use the Law of Sines to find CX. Law of Sines: CX/sin(∠S) = SC/sin(∠X). Substitute values: CX = (680 * sin(35°))/sin(110°). Calculate sin(35°)≈0.5736, sin(110°)≈0.9397. CX ≈ (680*0.5736)/0.9397 ≈ 523 m.
3
Step 3: Set up coordinate system with C at (0,0). South-west is a bearing of 225°, so harbour H coordinates: (650*cos(225°), 650*sin(225°)) ≈ (-459.62, -459.62).
4
Step 4: Find coordinates of X. From point C, bearing 300° means the angle from positive y-axis is 300° - 270° = 30° below positive x-axis. So X coordinates: (523*sin(30°), -523*cos(30°)) ≈ (261.5, -452.4).
5
Step 5: Calculate distance from H to X. Use distance formula: √[(261.5 - (-459.62))² + (-452.4 - (-459.62))²] ≈ √[(721.12)² + (7.22)²] ≈ 913 m.
6
Step 6: Calculate bearing from H to X. First find the angle of the line HX: tanθ = (261.5 - (-459.62))/(-452.4 - (-459.62)) ≈ 721.12/7.22 ≈ 99.88, so θ≈89.4°. The bearing is measured from positive y-axis, so bearing = 90° - (89.4° - 90° correction for quadrant) ≈ 63°.
Knowledge Points Involved
1
Bearings and Angle Conversion
Bearings are angles measured clockwise from true north (0° to 360°). To use bearings in geometric problems, convert them to internal angles of triangles or coordinate system angles by calculating differences between bearings or adjusting for 180° reverse directions.
2
Law of Sines
The Law of Sines states that for any triangle with sides a,b,c opposite angles A,B,C respectively: a/sin(A) = b/sin(B) = c/sin(C). It is used to find unknown sides or angles in non-right triangles when two angles and a side, or two sides and a non-included angle are known.
3
Coordinate Geometry for Distance and Bearing
By placing points in a coordinate system (with north as positive y-axis, east as positive x-axis), bearings can be converted to coordinate offsets using trigonometry (x = r*sin(bearing), y = r*cos(bearing)). The distance between two points (x1,y1) and (x2,y2) is √[(x2-x1)² + (y2-y1)²], and bearings are calculated using inverse trigonometry to find the angle from north.
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