Two ships leave a port at 9 a.m. One travels at a bearing of N 53° W at 12 miles per hour, and the other travels at a bearing of S 67° W at 19 miles per hour. Approximate how far apart they are at noon that day. (Round your answer to one decimal place.)
Two ships leave a port at 9 a.m. One travels at a bearing of N 53° W at 8 miles per hour, and the other travels at a bearing of S 67° W at 19 miles per hour. Approximate how far apart they are at noon that day
To find the distance between the two ships at noon, we need to calculate the displacement of each ship from the starting point.
Let's start with the ship traveling at a bearing of N 53° W:
- From 9 a.m. to noon, which is a duration of 3 hours, this ship travels at a speed of 12 miles per hour.
- The distance covered by the ship is 3 hours × 12 miles per hour = 36 miles.
Now let's move on to the ship traveling at a bearing of S 67° W:
- From 9 a.m. to noon, which is a duration of 3 hours, this ship travels at a speed of 19 miles per hour.
- The distance covered by the ship is 3 hours × 19 miles per hour = 57 miles.
To find the distance between the two ships, we can use the Pythagorean theorem:
Distance^2 = (36 miles)^2 + (57 miles)^2
Distance^2 = 1296 + 3249
Distance^2 = 4545
Taking the square root of both sides, we get:
Distance ≈ √4545 ≈ 67.5 miles
Therefore, the ships are approximately 67.5 miles apart at noon.
To find the distance between the two ships at noon, we need to determine how far each ship has traveled from the port. We can then use the distance formula to calculate the distance between the two points.
Let's start by calculating the distance traveled by each ship.
Ship A:
Speed = 12 miles per hour
Time traveled = 3 hours (from 9 a.m. to noon)
Distance traveled by Ship A = Speed × Time = 12 mph × 3 hours = 36 miles
Ship B:
Speed = 19 miles per hour
Time traveled = 3 hours (from 9 a.m. to noon)
Distance traveled by Ship B = Speed × Time = 19 mph × 3 hours = 57 miles
Now, we have the coordinates for each ship. Ship A is at point (0, 36) and Ship B is at point (0, -57). We can use the distance formula to calculate the distance between these two points.
Distance formula:
Distance = √[(x2 - x1)² + (y2 - y1)²]
In this case, the y-coordinates are opposite signs because one ship is traveling north while the other is traveling south.
Distance = √[(0 - 0)² + (-57 - 36)²]
Distance = √[0 + 8281]
Distance = √8281
Distance ≈ 91 miles
Therefore, the approximate distance between the two ships at noon is 91 miles.
My quick sketch shows a triangle with sides 36 and 57
with a contained angle between those sides of 60°
So you have a clear case of the cosine law
x^2 = 36^2 + 57^2 - 2(36)(57)cos60°
carry on