Two ships A and B left port (P) at the same time, with ship A moving due north while ship B was on a bearing of 060°. Two hours later A had covered a distance of 10km. B had covered 8km and was at a bearing of 150° from A. Calculate the distance separating the two ships after the two hours.
Makes no sense - over specified.
if you use the angles 150 and 60 you get a 30 , 30, 120 triangle and AB = OB = 8
however if you use B at (8 cos 60 , 8 sin 60) you get something else
If you draw the diagram, and label the points A and B for the ships and P for the port, you will see that you have a right triangle with
OB=8
OA=10
so, AB=6
Ships sail on headings, not bearings.
You did correctly use "bearing" when you gave the direction of B from A.
B, however, was still sailing on a heading of 60°
16
To calculate the distance separating the two ships after two hours, we can use the concept of vector addition.
Step 1: Determine the displacement of each ship after two hours.
Ship A: Ship A moved due north for two hours at a constant speed. The distance covered by Ship A in two hours is given as 10 km. Hence, the displacement of Ship A is a vector of magnitude 10 km and pointing due north.
Ship B: Ship B was on a bearing of 060° for two hours and covered a distance of 8 km. To determine the displacement of Ship B, we need to break down the distance covered into north and east components. Since the bearing is given as 060°, the angle with the X-axis is 60°. Using trigonometry, we can find the north and east components of the distance covered by Ship B.
North component = distance * sin(angle)
= 8 km * sin(60°)
= 8 km * √3/2
= 4√3 km
East component = distance * cos(angle)
= 8 km * cos(60°)
= 8 km * 1/2
= 4 km
Therefore, the displacement of Ship B is a vector with a north component of 4√3 km and an east component of 4 km.
Step 2: Calculate the net displacement between the two ships.
To find the net displacement between the two ships, we can subtract the displacement of Ship A from the displacement of Ship B.
Net displacement = Displacement of Ship B - Displacement of Ship A
Since Ship A moved due north and Ship B was at a bearing of 150° from A, the net displacement will be the difference between the north and east components of Ship B and the north component of Ship A.
North component of net displacement = North component of Ship B - North component of Ship A
= 4√3 km - 10 km
= -10 km + 4√3 km
= 4√3 km - 10 km
East component of net displacement = East component of Ship B
= 4 km
Hence, the net displacement between the two ships is a vector with a north component of 4√3 km - 10 km and an east component of 4 km.
Step 3: Calculate the distance between the two ships.
To calculate the distance between the two ships, we can use the Pythagorean theorem.
Distance = √(North component)^2 + (East component)^2
= √[(4√3 km - 10 km)^2 + (4 km)^2]
= √[(16 * 3 km - 80√3 km + 100 km^2) + 16 km^2]
= √[100 km^2 - 80√3 km + 16 km^2]
= √[116 km^2 - 80√3 km]
Now, you can use a calculator to simplify the expression and find the distance between the two ships after two hours.