man walks 2.0-kilometer due east and 2.0-kilometer due south. Then he runs 4.0 kilometers due west. Calculate the magnitude of the man's resultant displacement and the distance travelled.

To find the man's resultant displacement and the total distance traveled, we can break the problem into two parts: the distance traveled and the calculation of displacement. 1. **Total Distance Traveled**: - The man walks 2.0 kilometers east, 2.0 kilometers south, and then runs 4.0 kilometers west. - Total distance = Distance east + Distance south + Distance west - Total distance = 2.0 km + 2.0 km + 4.0 km = 8.0 kilometers. 2. **Resultant Displacement**: - To find the resultant displacement, we can use vector addition. - Starting point is at the origin (0,0). - After walking 2.0 km due east, his position is (2.0 km, 0). - After walking 2.0 km due south, his position is (2.0 km, -2.0 km). - After running 4.0 km due west, his new position is (2.0 km - 4.0 km, -2.0 km) = (-2.0 km, -2.0 km). Now, we can find the resultant displacement by calculating the distance from the origin to his final position (-2.0 km, -2.0 km). - The displacement can be calculated using the Pythagorean theorem: \[ \text{Displacement} = \sqrt{(x^2 + y^2)} \] where \(x = -2.0\) km and \(y = -2.0\) km. - Thus, \[ \text{Displacement} = \sqrt{(-2.0)^2 + (-2.0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2.83 \text{ km (approximately)}. \] **Final Results**: - Total distance traveled: 8.0 kilometers. - Magnitude of resultant displacement: approximately 2.83 kilometers.