Y is 60km away from X on a bearing of 135 degree. Z is 80km away from X on a bearing of 225 degree.
(a) find the distance of Z fromY?
(b) find the bearing of Z from y?
To find the distance of Z from Y, we need to use the concept of trigonometry and vector addition.
(a) Distance of Z from Y:
First, we need to find the coordinates of points X, Y, and Z on a two-dimensional plane.
Let's assume that point X is the origin (0, 0) of the coordinate system.
Point Y is located 60 km away from X on a bearing of 135 degrees. To find the coordinates of Y, we can use trigonometry. Using the angles in standard position, the x-coordinate can be found by calculating the horizontal component of the distance, which is given by 60 km * cos(135°), and the y-coordinate can be found by calculating the vertical component of the distance, which is given by 60 km * sin(135°).
x-coordinate of Y = 60 km * cos(135°) = -42.43 km
y-coordinate of Y = 60 km * sin(135°) = 42.43 km
Point Z is located 80 km away from X on a bearing of 225 degrees. Similarly, we can find the coordinates of Z using trigonometry.
x-coordinate of Z = 80 km * cos(225°) = 56.57 km
y-coordinate of Z = 80 km * sin(225°) = -56.57 km
Now that we have the coordinates of Y and Z, we can find the distance between them using the distance formula, which is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of Y and Z into the distance formula:
Distance of Z from Y = sqrt((56.57 km - (-42.43 km))^2 + (-56.57 km - 42.43 km)^2)
= sqrt(99.14^2 + (-99.14)^2)
= sqrt(9829.5396 + 9829.5396)
= sqrt(19659.0792)
= 139.95 km (rounded to two decimal places)
Therefore, the distance of Z from Y is approximately 139.95 km.
(b) Bearing of Z from Y:
To find the bearing of Z from Y, we need to use the angle between the line connecting Y and Z and the positive x-axis.
Bearing = arctan((y2 - y1)/(x2 - x1))
Substituting the coordinates of Y and Z into the bearing formula:
Bearing of Z from Y = arctan((-56.57 km - 42.43 km)/(56.57 km - (-42.43 km)))
= arctan((-99.14 km)/(99.14 km))
= arctan(-1)
= -45 degrees
Therefore, the bearing of Z from Y is -45 degrees. However, bearings are usually measured in the range from 0 to 360 degrees. In this case, we can add 360 degrees to the result to get the positive bearing:
Bearing of Z from Y = -45 degrees + 360 degrees
= 315 degrees
Therefore, the bearing of Z from Y is 315 degrees.