A man walks due west for 4km.He then changes direction and walks on a bearing of 197 degree until he is south west of his starting point.how far is he then from his starting point.

Answer 1

We can solve this problem using trigonometry and the cosine rule.

First, let's draw a diagram to represent the man's movements:

A
|\
4 | \ x
| \
B---C

The man starts at point A and walks west for 4km to point B. He then changes direction and walks on a bearing of 197 degrees (measured clockwise from due north) to point C, which is southwest of point A. We need to find the length of line segment AC (represented by x in the diagram).

To do this, we can use the cosine rule:

cos(C) = (a^2 + b^2 - c^2) / 2ab

where C is the angle opposite side c in the triangle ABC, and a, b, and c are the lengths of sides BC, AB, and AC respectively. We can use this formula to find the cosine of angle C, and then use the inverse cosine function (cos^-1) to find the value of angle C itself.

First, we need to find the lengths of sides AB and BC. We know that AB is 4km (given in the question), and we can find BC using trigonometry. The angle ABC is (180 - 197) = 17 degrees (since the sum of the angles in a triangle is always 180 degrees). To find BC, we can use the sine formula:

sin(B) = b / a

where B is the angle opposite side b, a is the hypotenuse (in this case, 4km), and b is the side we want to find (BC). Solving for b, we get:

b = a sin(B) = 4 sin(17) = 1.097 km (to 3 decimal places)

Now we can use the cosine rule to find the length of side AC:

cos(C) = (a^2 + b^2 - c^2) / 2ab

cos(C) = (4^2 + 1.097^2 - c^2) / (2 * 4 * 1.097)

cos(C) = (16 + 1.203 - c^2) / 8.776

c^2 = 17.836 - 8.776 cos(C)

We know that angle C is (180 - 17) = 163 degrees, since the angles in a triangle add up to 180 degrees. So we can substitute this value into the equation:

c^2 = 17.836 - 8.776 cos(163)

c^2 = 36.916

c = sqrt(36.916) = 6.082 km (to 3 decimal places)

Therefore, the man is 6.082 km from his starting point when he is southwest of it.

Answer 2

Sorry - the law of cosine will not help in this case, as we know only one side of the triangle.

Let's label
A = starting point
B = 4 km west of A
C = intersection point of lines with directions S45W of A and S17W of B
Then we have, in ∆ABC, AB=4, and C=28°, and we want to find AC
So now, the law of sines tells us that
AC/sin107° = 4/sin45°
AC = 5.41 km

Answer 3

You're correct, I apologize for my mistake. Using the Law of Sines as you did is the correct approach to solve this problem. Thank you for providing the correct solution!