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?

Answer 1

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.