You will be hiking to a lake with some of your friends by following the trails indicated on a map at the trailhead. The map says that you will travel 1.6 mi directly north, then \(2.2 \mathrm{mi}\) in a direction \(35^{\circ}\) east of north, then finally \(1.1 \mathrm{mi}\) in a direction \(15^{\circ}\) north of east. At the end of this hike, how far will you be from where you started, and what direction will you be from your starting point?

To find the displacement in each direction, we will use trigonometry for each step of the hike and add them up. In the first step, they hike directly North, making the displacement only in the North direction. Here, North displacement is `1.6 mi` and East displacement is `0 mi`. For the second step of the hike in a direction `35°` east of north, we can use the following trigonometry formulas: North displacement: \(2.2\,\mathrm{mi} \times \cos(35^{\circ})\), East displacement: \(2.2\,\mathrm{mi} \times \sin(35^{\circ})\). For the third step of the hike in a direction `15°` north of east, we use: North displacement: \(1.1\,\mathrm{mi} \times \sin(15^{\circ})\), East displacement: \(1.1\,\mathrm{mi} \times \cos(15^{\circ})\). Now, add the displacements in each direction separately.

Add the displacements found in step 1: Total North displacement: \(1.6\,\mathrm{mi} + 2.2\,\mathrm{mi} \times \cos(35^{\circ}) + 1.1\,\mathrm{mi} \times \sin(15^{\circ})\). Total East displacement: \(0 + 2.2\,\mathrm{mi} \times \sin(35^{\circ}) + 1.1\,\mathrm{mi} \times \cos(15^{\circ})\).

To find the distance from their starting point, use the Pythagorean theorem: Distance = \(\sqrt{\text{Total North displacement}^2 + \text{Total East displacement}^2}\). To find the direction, use the arctangent function in terms of displacements: Direction (angle) = \(\arctan\left(\frac{\text{Total East displacement}}{\text{Total North displacement}}\right)\). In the end, the hikers will be the calculated distance away from their starting point in the specified direction.