A man in search of his dog drives first 10 mi northeast, then \(12 \mathrm{mi}\) straight south, and finally \(8 \mathrm{mi}\) in a direction \(30^{\circ}\) north of west. What are the magnitude and direction of his resultant displacement?

First, we need to break down each direction into their individual components (East-North or West-South). Remember that East and North have positive values, while West and South have negative values. Trip 1: 10 mi northeast is 10 mi East and 10 mi North. East component: 10 mi North component: 10 mi Trip 2: 12 mi straight south. South component: -12 mi (since it's in the opposite direction of North) Trip 3: 8 mi in a direction 30° north of west. To find the components, we'll use trigonometric functions (sine and cosine) and the given angle. West component: 8 mi * cos(30°) = 6.93 mi (rounded to two decimal places) North component: 8 mi * sin(30°) = 4 mi Now we have the components for each trip: Trip 1: East=10 mi, North=10 mi Trip 2: South=-12 mi Trip 3: West=-6.93 mi, North=4 mi

Now we add up the components to find the total East-West and North-South components of the displacement. Total East component: 10 mi (from Trip 1) Total West component: -6.93 mi (from Trip 3) Total North component: 10 mi (from Trip 1) + 4 mi (from Trip 3) = 14 mi Total South component: -12 mi (from Trip 2) Then, combine the East and West components, as well as the North and South components. East-West component: 10 mi - 6.93 mi = 3.07 mi North-South component: 14 mi - 12 mi = 2 mi So, the man's total displacement has a horizontal component of 3.07 mi and a vertical component of 2 mi.

We can find the magnitude of his total displacement by using the Pythagorean theorem on the horizontal and vertical components. Magnitude = sqrt(East-West component^2 + North-South component^2) Magnitude = sqrt(3.07 mi^2 + 2 mi^2) Magnitude = 3.75 mi (rounded to two decimal places) So, the magnitude of the man's total displacement is approximately 3.75 mi.

For the direction of the displacement, let's find the angle between the displacement vector and the eastward direction. We can do this using the arctangent function (atan) on the ratio of the North-South component to the East-West component. Angle = atan(North-South component / East-West component) Angle = atan(2 mi / 3.07 mi) Angle = 33.1° (rounded to one decimal place) Since the man's total displacement is northeast, the angle with respect to the eastward direction is 33.1°. #Conclusion#The man's resultant displacement has a magnitude of 3.75 mi and is at an angle of 33.1° from the eastward direction (to the northeast).