A ship leaves port at 1:00 P.M. and travels S35°E at the rate of 28 mi/hr. Another ship leaves the same port at 1:30 P.M. and travels S20°W at 18 mi/hr. Approximately how far apart are the ships at 3:00 P.M.? (Round your answer to the nearest mile.)
I tried to solve it and I got 62 miles, but the answer is wrong. please HELP!
In my diagram, then angle between their path is 55°, and at 3:00 the first ship has gone 56 mi and the second ship has gone 27 miles.
Looks like a basic cosine law problem ...
x^2= 56^2 + 27^2 - 2(56)(27)cos55°
.....
x is not 62 miles, what did you get this time?
To solve this problem, we can find the distance each ship travels in the given time and then use the distance formula to find the distance between them.
Let's first calculate the distance the first ship travels from 1:00 P.M. to 3:00 P.M.:
The time difference is 2 hours (3:00 P.M. - 1:00 P.M.).
The speed of the first ship is 28 mi/hr, so the distance it travels is 28 mi/hr * 2 hrs = 56 miles.
Now let's calculate the distance the second ship travels from 1:30 P.M. to 3:00 P.M.:
The time difference is 1.5 hours (3:00 P.M. - 1:30 P.M.).
The speed of the second ship is 18 mi/hr, so the distance it travels is 18 mi/hr * 1.5 hrs = 27 miles.
To find the distance between the two ships, we can consider the triangle formed by the two ships and the distance between them. We can use the Law of Cosines to find the length of the third side of the triangle (the distance between the ships). The Law of Cosines states:
c² = a² + b² - 2ab * cos(C)
Here, c is the distance between the ships, a is the distance traveled by the first ship (56 miles), b is the distance traveled by the second ship (27 miles), and C is the angle between the paths of the two ships.
To find the angle C, we can subtract the given bearings:
S35°E - S20°W = S35°E + N20°E (converting to a common reference direction)
Since S and E are opposite directions on a compass, they cancel out, leaving N55°E.
Now we can calculate the distance between the ships:
c² = (56)² + (27)² - 2(56)(27) * cos(55°)
c² = 3136 + 729 - 3024 * cos(55°)
c² = 3865 - 3024 * cos(55°)
Using a calculator, we find that cos(55°) ≈ 0.5736. Plugging this into the equation:
c² ≈ 3865 - 3024 * 0.5736
c² ≈ 3865 - 1736.1536
c² ≈ 2128.8464
Now we take the square root of both sides to find the distance between the ships:
c ≈ √2128.8464
c ≈ 46.172 miles
Therefore, approximately, the ships are about 46 miles apart at 3:00 P.M.
To solve this problem, we can break it down into smaller steps.
Step 1: Determine the distance each ship has traveled from the port at 3:00 P.M.
The first ship left the port at 1:00 P.M. and has been traveling for a total of 2 hours. Since it travels at a rate of 28 miles per hour, the distance traveled by the first ship is 2 hours * 28 miles/hour = 56 miles.
The second ship left the port at 1:30 P.M. and has been traveling for a total of 1.5 hours. Since it travels at a rate of 18 miles per hour, the distance traveled by the second ship is 1.5 hours * 18 miles/hour = 27 miles.
Step 2: Determine the direction each ship is traveling.
The first ship is traveling S35°E, which means it is moving South 35 degrees East.
The second ship is traveling S20°W, which means it is moving South 20 degrees West.
Step 3: Calculate the horizontal and vertical components of each ship's movement.
For the first ship (S35°E):
Horizontal component = 56 miles * cos(35°) ≈ 45.74 miles
Vertical component = 56 miles * sin(35°) ≈ 31.91 miles
For the second ship (S20°W):
Horizontal component = 27 miles * cos(20°) ≈ 25.57 miles
Vertical component = -27 miles * sin(20°) ≈ -9.18 miles (negative sign indicates movement in the opposite direction)
Step 4: Calculate the horizontal and vertical distances between the ships.
Horizontal distance = absolute value (horizontal component of the first ship - horizontal component of the second ship) = |45.74 miles - 25.57 miles| = 20.17 miles
Vertical distance = absolute value (vertical component of the first ship - vertical component of the second ship) = |31.91 miles - (-9.18 miles)| = 41.09 miles
Step 5: Use the Pythagorean theorem to calculate the distance between the ships.
Distance = √((horizontal distance)^2 + (vertical distance)^2) = √((20.17 miles)^2 + (41.09 miles)^2) ≈ √(406.89 + 1687.88) ≈ √2094.77 ≈ 45.80 miles (rounded to the nearest mile)
Therefore, the approximate distance between the ships at 3:00 P.M. is 46 miles.