Dave walked from point A to point B due North East for a distance of 20m. He immediately changed direction and walked 50m from B to C on a bearing of 110 degrees.
i. Calculate the distance AC
ii. Find the bearing of C from A
1. d = 20m[45o] + 50m[110o]
X = 20*cos45 + 50*cos110 = -2.96 m.
Y = 20*sin45 + 50*sin110 = 61.1 m.
tanAr = Y/X = 61.1/-2.96 = -20.64189
Ar = -87.23o = Reference angle.
A = -87.23 + 180 = 92.8o
d = Y/sinA = 61.1/sin92.8 = 61.2 m.= AC.
2. Bearing = 92.8-90 = 2.8o West of North.
To solve this problem, we can use trigonometry and vector addition. Here's how:
i. To calculate the distance AC, we can break it down into two components: the north-south component and the east-west component.
1. The north-south component is the vertical displacement, which is 20m. Since Dave is walking in the northeast direction, this component is in the same direction as the bearing of 110 degrees.
2. The east-west component is the horizontal displacement, which we can find using trigonometry. The bearing of 110 degrees is measured clockwise from the north direction, so we need to find the cosine of the bearing angle to get the east-west component.
East-West Component (BE) = 50m * cos(110 degrees)
3. Now we can use the Pythagorean theorem to find the distance AC:
AC = √(North-South Component^2 + East-West Component^2)
= √(20m^2 + (50m * cos(110 degrees))^2)
You can use a calculator to compute the value.
ii. To find the bearing of point C from point A, we can use trigonometry again. We want to find the angle between the line AC and the north direction.
1. First, we find the angle between the north direction and the hypotenuse AC using inverse trigonometric functions:
Angle (θ) = tan^(-1)(North-South Component / East-West Component)
2. However, the bearing is measured clockwise from the north direction. So to get the bearing angle, we need to adjust it:
Bearing Angle = 90 degrees - Angle (θ)
Again, you can use a calculator to compute the value.
By following these steps, you should be able to calculate the distance AC and the bearing of point C from point A.