A ship sails 5 km from a port A on a bearing of 85 degrees and then 6 km on a bearing of 50 degrees. Calculate the distance and bearing from A.
calculate the horizontal and vertical replacements of each trip
first trip:
x/5 = cos85 , x = 5cos85
y/5 = sin85 , y = 5sin85
2nd trip:
x/6 = cos50 , x = 6cos50
y/6 = sin50 , y = 6sin50
total horizontal displacement
= 5cos85 + 6cos50 = appr 4.2925
total vertical displacement
= 5sin85 + 6sin50 = appr 9.5772
now use Pythagoras ... to get
√(4.2925^2 + 9.5772) = 40.4952
or
just add the vectors
((5cos85,5sin50) + (6cos50,6sin50) and take their magnitude.
(the actual calculations are the same as above)
THANK YOU
Are we worried whether we use compass headings where North = 0 degrees, rather than East, and angles are measured clockwise?
To calculate the distance and bearing from point A, we can use the method of vectors.
Step 1: Draw a diagram
First, draw a diagram with point A representing the port, and mark the distances and bearings given in the question. Label the endpoint of the ship's journey as point B.
Step 2: Break down the vectors
Break down the vectors into their horizontal (x) and vertical (y) components. This can be done using trigonometry.
For the first leg of the journey:
- Distance = 5 km
- Bearing = 85 degrees
To find the horizontal component (x1) of the first leg, use the formula:
x1 = distance * cos(bearing)
Substituting the values:
x1 = 5 km * cos(85 degrees)
To find the vertical component (y1) of the first leg, use the formula:
y1 = distance * sin(bearing)
Substituting the values:
y1 = 5 km * sin(85 degrees)
Similarly, for the second leg of the journey:
- Distance = 6 km
- Bearing = 50 degrees
Find the horizontal component (x2) and the vertical component (y2) using the same formulas.
Step 3: Determine the total x and y components
To find the total horizontal and vertical components, add the corresponding components of both legs:
xtotal = x1 + x2
ytotal = y1 + y2
Step 4: Calculate the distance and bearing from point A to point B
The distance from point A to point B is given by the formula:
distance = sqrt(xtotal^2 + ytotal^2)
The bearing from point A to point B can be found using the inverse tangent function (arctan). The formula is:
bearing = arctan(ytotal/xtotal)
Substitute the values of xtotal and ytotal into these formulas to calculate the distance and bearing.
Note: The angles used in the trigonometric functions are typically in radians, so it's important to convert between degrees and radians if necessary.
By following these steps, you can calculate the distance (in km) and bearing (in degrees) from point A.