A ship leaves port and travels 21km on a bearing of 032 degrees and then 45km on a bearing of 287 degrees. A. calculate it distance from the port. B. calculate the bearing of the port from the ship
21km on a HEADING of 032 degrees
distance north = 21 cos 32 = 17.8
distance West = -21 sin 32 (really east so negative) = -11.1
45km on a HEADING of 287 degrees (73 deg west of north)
distance north = 45 cos 73 = 13.2
distance west = 45 sin73 = 43.0
total north = 31
total west = 31.9
distance =sqrt (31^2 + 31.9^2) = 44.5
tan angle west of north = 31.9/31 = 1.03
so angle w of n = 45.8
360 - 45.8 = 314 deg bearing from start
A. To calculate the distance from the port, we can use the Pythagorean theorem. Let's break down the ship's journey into two legs:
Leg 1: Traveling 21km on a bearing of 032 degrees.
Leg 2: Traveling 45km on a bearing of 287 degrees.
Using Leg 1:
The horizontal distance traveled is given by:
Horizontal distance = Leg 1 distance * cos(Leg 1 angle)
Horizontal distance = 21km * cos(32°)
Using Leg 2:
The horizontal distance traveled is given by:
Horizontal distance = Leg 2 distance * cos(Leg 2 angle)
Horizontal distance = 45km * cos(287°)
Now we can calculate the total horizontal distance covered by the ship:
Total horizontal distance = Horizontal distance (Leg 1) + Horizontal distance (Leg 2)
B. To calculate the bearing of the port from the ship, we need to find the angle formed between the line connecting the ship to the port and the true North direction. This can be calculated using trigonometry.
Let's calculate the values step-by-step:
A. Distance from the port:
Using Leg 1:
Horizontal distance = 21km * cos(32°)
Horizontal distance ≈ 17.8388 km
Using Leg 2:
Horizontal distance = 45km * cos(287°)
Horizontal distance ≈ 36.2202 km
Total horizontal distance = 17.8388 km + 36.2202 km
Total horizontal distance ≈ 54.059 km
To calculate the distance from the port, we can use the Pythagorean theorem. The distance from the port is the hypotenuse of the right-angled triangle formed by the horizontal distance and the vertical distance traveled.
Using Leg 1:
Vertical distance = Leg 1 distance * sin(Leg 1 angle)
Vertical distance = 21km * sin(32°)
Vertical distance ≈ 11.068 km
Using Leg 2:
Vertical distance = Leg 2 distance * sin(Leg 2 angle)
Vertical distance = 45km * sin(287°)
Vertical distance ≈ -25.471 km (negative sign indicates direction)
To find the total vertical distance, we add the vertical distances from both legs:
Total vertical distance = Vertical distance (Leg 1) + Vertical distance (Leg 2)
Total vertical distance ≈ 11.068 km - 25.471 km
Total vertical distance ≈ -14.403 km
Now, using the Pythagorean theorem:
Distance from the port = sqrt((Total horizontal distance)^2 + (Total vertical distance)^2)
Distance from the port = sqrt((54.059 km)^2 + (-14.403 km)^2)
Distance from the port ≈ sqrt(2925.472181 km^2 + 207.096609 km^2)
Distance from the port ≈ sqrt(3132.56879 km^2)
Distance from the port ≈ 55.992 km
Therefore, the distance from the port is approximately 55.992 km.
B. Bearing of the port from the ship:
To find the bearing of the port from the ship, we can use the inverse tangent function (arctan). The bearing is taken from the direction of the ship to the port in a clockwise direction.
Bearing of the port from the ship = arctan(Vertical distance / Horizontal distance)
Bearing of the port from the ship ≈ arctan(-14.403 km / 54.059 km)
Using a calculator or trigonometric tables, we find that arctan(-14.403 / 54.059) ≈ -15.123 degrees.
The bearing of the port from the ship is approximately 15.123 degrees clockwise from the ship's direction.
To solve this problem, we can use the concepts of vector addition and trigonometry.
A. To calculate the distance from the port, we need to find the resultant of the two displacements. We can represent each displacement as a vector with magnitude and direction.
First, let's convert the given bearings into angles measured counterclockwise from the positive x-axis.
For the first displacement, bearing 032 degrees corresponds to an angle of (360 - 32) = 328 degrees. Similarly, for the second displacement, bearing 287 degrees corresponds to (360 - 287) = 73 degrees.
Now, we can find the x and y components of each displacement. The x-component can be calculated using the formula cos(angle) * magnitude, while the y-component can be calculated using sin(angle) * magnitude.
For the first displacement:
x1 = cos(328 degrees) * 21 km
y1 = sin(328 degrees) * 21 km
For the second displacement:
x2 = cos(73 degrees) * 45 km
y2 = sin(73 degrees) * 45 km
Next, we can add up the x and y components separately to find the resultant displacement.
Rx = x1 + x2
Ry = y1 + y2
Finally, we can use the Pythagorean theorem to find the distance of the ship from the port:
Distance = √(Rx^2 + Ry^2)
B. To calculate the bearing of the port from the ship, we can use the trigonometric inverse functions to find the angle between the resultant displacement vector and the positive x-axis.
Bearing = 360 degrees - arctan(Ry/Rx)
Now, let's calculate the values:
First, convert the angles to radians:
Angle1 = 328 degrees * π/180 radians
Angle2 = 73 degrees * π/180 radians
Calculate the x and y components:
x1 = cos(Angle1) * 21 km
y1 = sin(Angle1) * 21 km
x2 = cos(Angle2) * 45 km
y2 = sin(Angle2) * 45 km
Calculate the resultant displacement:
Rx = x1 + x2
Ry = y1 + y2
Calculate the distance:
Distance = √(Rx^2 + Ry^2)
Calculate the bearing:
Bearing = 360 - arctan(Ry/Rx) * 180/π degrees
Plug in the values to get the final answer!