a hiker travels 3 km due east then 2km on a bearing of 110° how far south is the hiker from the starting point
due east is 90º
110º is 20º south of east
distance south of the start is ... 2 km * sin(20º)
Draw a horizontal line.
Mark start point as A.
Mark the point after 3 km as B.
At that point B, draw an angle θ = 110 ° with respect to the horizontal.
Mark the point after 2 km as C.
∠ θ = 110° = ∠ ABC
∠ is the mark for angle
Mark the length AB as a.
a = 3 km
Mark the length BC as b.
b = 2 km
Mark the length AC as c.
Now law of cosines:
c² = a² + b² - 2 ∙ a ∙ b ∙ cos θ
c² = a² + b² - 2 ∙ a ∙ b ∙ cos 110°
c² = 3² + 2² - 2 ∙ 3 ∙ 2 ∙ cos 110°
c² = 9 + 4 - 12 ∙ cos 110°
c² = 13 - 12 ∙ ( - 0.342020143 )
c² = 13 + 4.104241716
c² = 17.104241716
c = √17.104241716
c = 4.1357274712 km
"how far south is the hiker from the starting point"
means that you only want the y co-ordinate of the resulting vector
(3cos0 , 3sin0) + (2cos(-20), 2sin(-20))
= (3,0) + (1.8794, -.684)
= (4.8794, -.684)
So the southward position from his starting point is .684 km
If we want the distance from his starting points we get
√(4.8794^2 + (-.684)^2) = 4.927 km
If you change the angle in Bosnian's solution to 160°, we get the same distance answer.
To determine how far south the hiker is from the starting point, we can break down the movement into its east-west and north-south components.
First, let's determine the east-west displacement. The hiker travels 3 km due east, so the east-west component is 3 km.
Now, let's find the north-south component. The hiker travels 2 km on a bearing of 110°. To determine the north-south displacement, we need to find the component of this distance that is in the south direction.
To do that, we can use trigonometry. We'll use the sine function because it relates the lengths of the sides of a right triangle to the angle opposite the side we want to find.
The angle between the 110° bearing and the south direction is 180° - 110° = 70°. Now, we can use the sine function to find the north-south component:
sin(70°) = (north-south displacement) / (2 km)
Rearranging the formula, we find:
(north-south displacement) = sin(70°) * (2 km)
Calculating this, we get:
(north-south displacement) ≈ 1.90 km
Therefore, the hiker is approximately 1.90 km south of the starting point.