A coast guard helicopter hovers between an island and a damaged sailboat. How would you calculate the direct distance between the helicopter and the sailboat? From the island, the angle of elevation to the helicopter is 73 degrees. From the helicopter, the island and the sailboat are 40 degrees apart. A police rescue boat heading toward the sailboat is 800 m away from the scene of the accident. From this position, the angle between the island and the sailboat is 35 degrees. At the same moment, an observer on the island notices that the sailboat and the police rescue boat are 68 degrees apart. Explain how you would calculate the straight-line distance, to the nearest metre, from the helicopter to the sailboat. Justify your reasoning with calculations.
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Make a triangle (1) between the sailboat, the island and the rescue boat, a triangle (2) between the sailboat the island and the helicopter, a triangle (3) from the island to the helicopter by horizontal, vertical and direct distance and a similar triangle (4) from the sailboat to the helicopter.
Triangle (1)
The angle between the island and the sailboat from the point of view of the rescue boat is A=35 and the angle between the resuce boat and the sailboat from the point of view of the island is B = 68. The third angle (the angle between the island and the rescue boat from the point of view of the sailboat) is then
C = 180 - (35+68) = 213`^@`
The distance ` `between the rescue boat and the sailboat is 800m. Using the sine rule, the distance `d` between the island satisfies
`sinB/800 = sinA/d`
Therefore `d = 800(sinA)/sinB = 800(sin35)/sin68 = 494.90m`
Triangle (2)
The angle between the sailboat and the island from the point of view of the helicopter is 40. and the distance between them is `d`. Call the other two angles ` `E and F. and the other two lengths `e` and `f` where `f` is the horizontal distance between the island and the helicopter.
Now `sin40/800 = sinE/e = sinF/f`
So `e = f((sinE)/sinF)` and `E + F = 180-40 = 140`
Triangle (3)
The elevation of the helicopter is 73 from the point of view of the island. The vertical height `g` of the helicopter satisfies
`(sin73/g) = sin(90-73)/f`
`implies g = f(sin73)/sin(17) = 0.704f`
Triangle (4)
We now have that the direct distance `h` between the helicopter and the sailboat satisfies
`h^2 = e^2 +g^2 = f^2(0.704^2 + ((sinE)/(sinF))^2)`
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