A man walks due west for 4km. He then changes direction and walks on a bearing of 179 until he is south-west of his starting point.How far is he then from his starting point?
I dont understand
use the law of sines. The distance z is
z/sin91 = 4/sin44
To find the distance from the man's starting point, we can use the concept of vectors.
Let's assume the man's starting point is called point A.
- He walks due west for 4km from point A. This means he moves 4km in the west direction. We can represent this vector as -4i since the west direction is opposite to the positive x-axis.
- He then changes direction and walks on a bearing of 179 degrees. This means the angle between his new direction and the positive x-axis is 179 degrees. However, since he is southwest of his starting point, the new direction will be in the second quadrant. Therefore, the angle measured from the positive x-axis counterclockwise will be 179 + 180 = 359 degrees.
- The distance he walks in this new direction is unknown, so let's call this distance d. The vector representation for this distance will be d * cos(359)i + d * sin(359)j, where i and j are unit vectors pointing in the positive x and y directions respectively.
We can set up an equation based on the displacement vector to find the value of d.
Displacement vector = -4i + d * cos(359)i + d * sin(359)j
The displacement vector represents the final position of the man relative to his starting point. If he is exactly southwest of his starting point, the x-component of the displacement vector should be 0, and the y-component should also be 0.
Setting up the equation:
-4 + d * cos(359) = 0 (x-component)
0 + d * sin(359) = 0 (y-component)
From the equations above, we can solve for d to find the distance from the starting point.
To find the distance the man is from his starting point after changing direction, we can break down his movements into components.
First, he walks due west for 4km, which means he moves directly along the x-axis in the negative direction.
Next, he changes direction and walks on a bearing of 179 degrees. To figure out the components of this movement, we can use trigonometry.
Since the bearing is 179 degrees and we know that he started walking west, we can determine that this new bearing corresponds to a direction that is almost directly south, but just slightly west of south.
The angle between due south and his new direction is 180 - 179 = 1 degree.
Let's assume that the distance he traveled along this new direction is 'd' km.
Using trigonometry, we know that the sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is 'd' km and the hypotenuse is the distance from the starting point to his new position. So, we have:
sin(1 degree) = (d km) / hypotenuse
To find the hypotenuse, we rearrange the equation:
hypotenuse = (d km) / sin(1 degree)
Now, we need to determine the value of 'd' by considering the bearing. Since the bearing is slightly west of due south, we can use the cosine of the angle to find the adjacent side.
cos(1 degree) = adjacent side / hypotenuse
Rearranging the equation, we can solve for the adjacent side:
adjacent side = cos(1 degree) * hypotenuse
We know that the distance traveled along the new direction is 'd' km, so the adjacent side is also equal to 4km.
Therefore, we can set up the following equation:
4km = cos(1 degree) * ((d km) / sin(1 degree))
To find the value of 'd', we can now solve for it:
d km = (4km * sin(1 degree)) / cos(1 degree)
Using a calculator, we can evaluate sin(1 degree) and cos(1 degree) to find their decimal values. We get:
d km ≈ (4km * 0.017452) / 0.999848
Clearly,
d ≈ (4 * 0.017452) / 0.999848
d ≈ 0.069808 / 0.999848
d ≈ 0.069823 km
So, the man is approximately 0.069823 km away from his starting point after changing direction.