A man walks due west for 4km.He then changes direction and walks on a bearing of 197 until he is South West of his starting point.How far is he then from his starting point?
To solve this problem, we can use trigonometry and the properties of right triangles.
Let's call the point where the man is after walking due west for 4 km as point A.
Let's call the point where the man is after changing direction and walking on a bearing of 197 as point B.
We need to find the distance between point A and point B.
First, let's consider the triangle formed by point A, point B, and point C, where point C is the starting point.
Since the man walks due west for 4 km, the distance between point A and point C is 4 km.
We know that the bearing of 197 is in the Southwest direction, which means that angle BAC, or ∠BAC, is 180° - 197° = 17° (angle A is 180°, angle C is the 197° bearing, so angle BAC is the difference).
Now, let's use trigonometry to find the distance between point A and point B. We can use the sine function since we have the opposite (4 km) and the hypotenuse (the distance between point A and B, which we'll call x).
Considering the right triangle ABC, we can write:
sin(17°) = opposite/hypotenuse
sin(17°) = 4/x
To solve for x, we rearrange the equation:
x = 4/sin(17°)
x ≈ 14.85 km
Therefore, the man is approximately 14.85 km from his starting point after changing direction and walking on a bearing of 197.