Let's approach the problem using trigonometry to find the angle of the triangle formed by the two hikers and their respective displacements.
The first hiker travels N35W, which means the angle between his displacement and the north direction is 35 degrees. Similarly, the second hiker travels S25W, so the angle between his displacement and the south direction is 25 degrees.
To find the angle between the two hikers, we can consider the angle formed at the point where their displacements meet. Since one displacement is towards the north and the other is towards the south, the angle formed at their meeting point is the sum of the angles mentioned above:
Angle = 35 degrees + 25 degrees = 60 degrees
Now, we have an isosceles triangle with two sides of equal length: 5 km and 8 km. The included angle between these sides is 60 degrees.
To find the distance between the two hikers (the remaining side of the triangle), we can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(angle)
Substituting the values:
c^2 = 5^2 + 8^2 - 2 * 5 * 8 * cos(60)
Simplifying the equation and calculating:
c^2 = 25 + 64 - 80 * cos(60)
c^2 = 89 - 80 * (1/2)
c^2 = 89 - 40
c^2 = 49
Taking the square root of both sides:
c = sqrt(49)
c = 7 km
Therefore, the two hikers are approximately 7 km apart.
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