What is the displacement for the couple's hiking trip? Give your answer as an ordered pair, with magnitude first, followed by a comma, then followed by the direction. Give the direction in terms of an angle measured CCW from the Easterly direction.

displacement = ( 4.045 km , 72 degrees )

Explanation:

Given:

- The radius if the circular path r = 2.5 km

- Couple hikes 70 % of the circular track.

Find:

What is the displacement for the couple's hiking trip? Give the direction in terms of an angle measured CCW from the Easterly direction.

Solution:

- The start point of the journey is west and due south in a circular direction.

- First we will determine the 70 % of the circumference of the circular path traveled. Using the arc length S formula:

                                       S = 0.70* ( 2*pi*r) = 7/5 pi*r

- The arc length calculated above is the distance traveled. However, we are interested in the "displacement" which is the shortest distance from start point to 70% of the track.

- The total amount of angle swept between the two points in radians is:

                                        S = r*Q

                                        Q = S / r = 7/5*pi

- The remaining angle angle to complete the full circle:

                                        Q' = 2*p - Q = 2*p -7/5*pi

                                        Q' = 0.6*pi.

- The displacement between two points can be calculated using sine rule as follows:

                     sin(Q') / displacement = sin (0.2*pi) / r

                     displacement = r*sin(Q') / sin (0.2*pi)

                     displacement = 2.5*sin(0.6*pi) / sin (0.2*pi)

                    displacement = 4.045 km

- The position with respect to the East direction is:

                                        pi - Q' = pi - 0.6*pi = 0.4*pi

Convert into degrees:               0.4*pi * 180 /pi = 72 degrees