Two ships leave a port, sailing 18 km/h and 26 km/h. Their angle between their directions oftravel from the port is 39° . How far part are the ships to the nearest km after 2 hours?15) _____________________________3 marks
Question
Two ships leave a port, sailing 18 km/h and 26 km/h. Their angle between their directions oftravel from the port is 39° . How far part are the ships to the nearest km after 2 hours?15) _____________________________3 marks
Solution
To solve this problem, we can use the law of cosines. The law of cosines is a formula used in geometry to find a side of a triangle when the lengths of the other two sides and the angle between them are known.
Here are the steps:
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First, calculate how far each ship has traveled. This is done by multiplying the speed of the ship by the time it has been traveling.
For the first ship: 18 km/h * 2 hours = 36 km For the second ship: 26 km/h * 2 hours = 52 km
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Now, we can use the law of cosines to find the distance between the two ships. The formula is:
c² = a² + b² - 2ab*cos(C)
where: a and b are the sides of the triangle (the distances each ship has traveled) C is the angle between the sides (the angle between the ships)
So, we have:
c² = 36² + 52² - 23652*cos(39°)
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Solve the equation:
c² = 1296 + 2704 - 23652*0.7771 (cos(39°) = 0.7771) c² = 4000 - 3007.44 c² = 992.56
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Take the square root of both sides to solve for c (the distance between the ships):
c = sqrt(992.56) = 31.5 km
So, to the nearest km, the ships are 32 km apart after 2 hours.
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