a car travels N28 degrees 20 minutes for 86 km and then N61 degrees 40 minutes W for 124 km what is its distance and bearing from the sticking point
someone answer pls
I made a sketch and marked all the angles
Let r be the distance between start and ending point.
the angle between the two trips is 90+28°20'+28°20' = 146° 40'
by the cosine law:
r^2 = 86^2 + 124^2 - 2(86)(124)cos146°40'
= 40591.28405
r = √...
= 201.477874.. km
angle of the triangle at start position:
sinØ/124 = sin146°40'/201.477874
sinØ = .3382...
Ø = 19.768° or 19°46'
so the angle using your notation
= 19°46' + 28°20' = 48°6
or N48°6'
using vectors:
|r| = (86cos28°20' , 86sin28°20') + (124cos61°40',124sin61°40')
= (75.6973.. , 40.8156) + (58.8504 , 109.14497)
= (134.54776.. , 149.9606...)
= √(134.54776..^2 + 149.9606..^2)
= 201.472787 , should have been exactly the same as above, probably typed in a decimal digit wrong
(I will let you find it, if you feel like it )
angle:
tan(angle) = 149.96.../134.54... = 1.11455..
angle = 48.1° = 48° 5'
again, that angle should be the same as above, I don't know where my arithmetic error is
(there is no error in the method)
To find the distance and bearing from the starting point, we can break down the problem into two components:
1. Calculate the coordinates of the ending point using the given distances and angles.
2. Find the distance and bearing between the starting point and the ending point.
Let's start with the first component:
1. Calculate the coordinates of the ending point:
a) For the first part of the journey, the car travels N28 degrees 20 minutes for 86 km. This tells us that the car moves 86 km in the direction of North 28 degrees 20 minutes.
b) To convert the angle to decimal degrees, we use the formula:
decimal degrees = degrees + (minutes / 60)
In this case, we have 28 degrees + (20 minutes / 60) = 28.3333 degrees.
c) Since the car travels towards the north, the latitude of the ending point will increase by 28.3333 degrees, and the longitude will remain the same.
Therefore, the coordinates of the ending point after the first part of the journey are:
Latitude: +28.3333 degrees
Longitude: No change from the starting point
d) For the second part of the journey, the car travels N61 degrees 40 minutes W for 124 km. This tells us that the car moves 124 km in the direction of North 61 degrees 40 minutes West.
e) Converting the angle to decimal degrees, we have:
decimal degrees = degrees + (minutes / 60)
In this case, we have 61 degrees + (40 minutes / 60) = 61.6667 degrees.
f) Since the car travels towards the northwest, the latitude of the ending point will increase by 61.6667 degrees, and the longitude will decrease by 90 degrees (as W is 90 degrees west).
Therefore, the coordinates of the ending point after the second part of the journey are:
Latitude: +28.3333 degrees + 61.6667 degrees = 90 degrees
Longitude: No change from the starting point - 90 degrees = 0 degrees West
Now that we have the coordinates of the ending point, we can move on to the second component:
2. Find the distance and bearing between the starting point and the ending point:
a) The distance between two points on the Earth's surface can be calculated using the haversine formula or the Vincenty formula. However, since the given distances are relatively small, we can use the Pythagorean theorem to calculate the approximate distance.
Distance = sqrt((latitude difference)^2 + (longitude difference)^2)
b) In this case, the latitude difference is 90 degrees - 0 degrees = 90 degrees, and the longitude difference is 0 degrees - 0 degrees = 0 degrees.
Therefore, the distance between the starting and ending points is:
Distance = sqrt((90 degrees)^2 + (0 degrees)^2) = sqrt(90^2) = 90 km
c) To calculate the bearing, we can use the inverse tangent (atan2) function:
Bearing = atan2(latitude difference, longitude difference)
d) In this case, the latitude difference is 90 - 0 = 90 degrees, and the longitude difference is 0 - 0 = 0 degrees.
Therefore, the bearing between the starting and ending points is:
Bearing = atan2(90 degrees, 0 degrees) = 90 degrees
So, the distance from the starting point is approximately 90 km, and the bearing from the starting point is 90 degrees.