In an extreme adventure triathalon, participants swim 1.7 Km from a dock to one end of an island, run 1.5 Km due north along a length of the island, and then kayak back to the dock.from the dock, the angel between the lines of sight to the ends of the island measures 15 degrees. How long is the kayak leg of the race?
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Well, you could do it by sine law.
Let the angle between the 1.5km run and the kayak leg be A,
then (sinA)/1.7 = (sin15°)/1.5
hence find A (two possible values, the acute value and its supplement), hence the angle between the 1.7 and 1.5 legs, and you could choose between cosine rule and sine rule to find the kayak leg.
However, I'd use the cosine rule from the beginning, letting x km be the kayak leg, then
1.5² = x² + 1.7² - 2*1.7x cos(15°)
i.e.
x² - 3.4 x cos(15°) + 1.7² - 1.5² = 0
(constant term 1.7² - 1.5² = 0.64)
x = [3.4 cos(15°) ±√((3.4 cos(15°))² - 4*0.64)] / 2
.. = 3.0761 or 0.2081
Common sense suggests the larger answer, rounded to 3.1km since the other distances are given to one decimal place.
With the sine rule method,
sinA = 1.7(sin15°)/1.5
...... = 0.293328
hence
A = 17.06° or 162.94°
The first one gives an angle of (180 - 17.06 - 15)° = 147.94° between the swim and run,
and then
x/sin(147.94°) = 1.5/sin(15°)
hence x = 3.076 as before;
The second answer means the run is in very roughly the opposite direction to the swim, and the angle between those two sides of the triangle is only 2.04°; the run has wiped out much of the distance covered by the swim, and the only distance to kayak is
x = 1.5 sin(2.04°)/sin(15°)
.. = 0.206, which seems unrealistic.
Let the angle between the 1.5km run and the kayak leg be A,
then (sinA)/1.7 = (sin15°)/1.5
hence find A (two possible values, the acute value and its supplement), hence the angle between the 1.7 and 1.5 legs, and you could choose between cosine rule and sine rule to find the kayak leg.
However, I'd use the cosine rule from the beginning, letting x km be the kayak leg, then
1.5² = x² + 1.7² - 2*1.7x cos(15°)
i.e.
x² - 3.4 x cos(15°) + 1.7² - 1.5² = 0
(constant term 1.7² - 1.5² = 0.64)
x = [3.4 cos(15°) ±√((3.4 cos(15°))² - 4*0.64)] / 2
.. = 3.0761 or 0.2081
Common sense suggests the larger answer, rounded to 3.1km since the other distances are given to one decimal place.
With the sine rule method,
sinA = 1.7(sin15°)/1.5
...... = 0.293328
hence
A = 17.06° or 162.94°
The first one gives an angle of (180 - 17.06 - 15)° = 147.94° between the swim and run,
and then
x/sin(147.94°) = 1.5/sin(15°)
hence x = 3.076 as before;
The second answer means the run is in very roughly the opposite direction to the swim, and the angle between those two sides of the triangle is only 2.04°; the run has wiped out much of the distance covered by the swim, and the only distance to kayak is
x = 1.5 sin(2.04°)/sin(15°)
.. = 0.206, which seems unrealistic.