a girl walks from a point A to a point B,20Km Away From A In Northeast Direction And WalkS To a point C on the bearing of 125degree,she is 80km away from the starting point,find her bearing from the starting point
Northeast ----> 45°
After you make your sketch you will have triangle ABC, where
Angle B = 100°, AB = 20 and AC = 89
you have to find angle A
By the sine law, first find angle C
SinC/20 = sin100/80
Angle C = ...
angle A = 180 - 100 - C
From there use your diagram to find the "bearing"
Given: AB = 20km[45o],
BC = BC[125o],
AC = 80km[Xo],
Draw triangle ABC:
B = 125 - 45 = 80o,
C = 125 - 90 = 35o,
A = 180 - (80+35) = 65o.
Law of sine:
BC/sin65 = 80/sin80,
BC = 74 km.
AC = AB + BC = 20km[45o] + 74km[125o],
AC = (20*sin45+74*sin125) + (20*Cos45+74*Cos125)I = 74.8 - 28.3i,
AC = 80km[-69.3o] = 80km[69.3o] E. of S. = 80km[110.7o] CW(bearing).
To find the girl's bearing from the starting point, we can use trigonometry and the concept of bearings.
Here's a step-by-step explanation of how to find the girl's bearing:
1. Draw a diagram: Draw a diagram that represents the situation described. Place point A as the starting point and point B as the destination, which is 20 km away from A in the northeast direction. Also, mark point C, which is 80 km away from the starting point on a bearing of 125 degrees.
2. Determine the angle: To find the girl's bearing from the starting point, we need to determine the angle between the north direction and the line from the starting point (A) to the final point (C). In this case, the angle would be the sum of the northeast angle and the bearing angle.
3. Compute the northeast angle: Considering that northeast is a combination of north and east, it forms a right angle triangle. Since we have the hypotenuse (20 km), we can use trigonometry to find the sides of the triangle. The north component of the northeast direction would be the opposite side, and the east component would be the adjacent side.
Using Pythagoras' theorem, we can calculate the north component (opposite side):
North component = sqrt((20 km)^2 - (20 km/√2)^2)
Given that sqrt(2) is approximately 1.414, we can simplify the equation:
North component = sqrt((20 km)^2 - (20 km)^2 / 1.414^2)
North component = sqrt(400 km^2 - 282.84 km^2)
North component = sqrt(117.16 km^2)
North component ≈ 10.81 km
Now, let's compute the east component (adjacent side):
East component = 20 km / √2
East component ≈ 14.14 km
4. Find the angle: With the north and east components, we can find the angle using trigonometry. The tangent function relates the angle to the opposite and adjacent sides:
Tan(angle) = opposite/adjacent
Tan(angle) = 10.81 km / 14.14 km
Taking the inverse tangent (arctan) of both sides, we get:
Angle = Arctan(10.81 km / 14.14 km)
Angle ≈ 38.33 degrees
5. Determine the bearing: To find the girl's bearing from the starting point, we need to add the angle calculated in step 4 to the bearing angle given (125 degrees in this case).
Bearing = 125 degrees + 38.33 degrees
Bearing ≈ 163.33 degrees
So, the girl's bearing from the starting point is approximately 163.33 degrees.