Jake is on the Appalachian Trial hiking. He hikes 8 miles due north and then turns and hikes an additional 10 miles at a bearing of N 67°W at which point he has reached a vista, eats lunch, and decides to bushwhack directly back to the trailhead. How far is he now from the trailhead? What bearing must he take to get back to the trailhead?
To find out how far Jake is from the trailhead and the bearing he should take to return, we can break down his journey into smaller steps.
Step 1: Hiking 8 miles due north
This means Jake has moved directly in the north direction for 8 miles.
Step 2: Hiking an additional 10 miles at a bearing of N 67°W
To understand this, we need to visualize the bearing.
The bearing of N 67°W means that Jake is moving 67° westward (leftward) from north. So, starting from due north (upwards direction), he turns towards his left at a 67° angle and continues walking for 10 miles.
Now, let's break down the 10 miles into northward and westward components.
Northward component: To find this, we need to determine the vertical displacement of Jake from his original position due to the 10-mile hike at the given bearing. We can calculate this using trigonometry.
The northward component can be calculated as: northward distance = distance * cos(bearing angle)
northward distance = 10 miles * cos(67°) ≈ 3.88 miles
Westward component: To find this, we need to determine the horizontal displacement of Jake from his original position due to the 10-mile hike at the given bearing.
The westward component can be calculated as: westward distance = distance * sin(bearing angle)
westward distance = 10 miles * sin(67°) ≈ 9.15 miles
Step 3: Bushwhacking directly back to the trailhead
To find the distance and bearing Jake needs to follow to return to the trailhead, we can consider the displacements from the previous steps.
Distance from the trailhead = 8 miles (north) + 3.88 miles (northward displacement) ≈ 11.88 miles
For the bearing back to the trailhead, we can use trigonometry to find the angle between Jake's final position and the trailhead.
tan(bearing angle) = northward displacement / westward displacement
bearing angle = arctan(northward displacement / westward displacement)
bearing angle = arctan(3.88 miles / 9.15 miles) ≈ 23.9°
Therefore, Jake is approximately 11.88 miles from the trailhead, and he needs to take a bearing of approximately N 23.9°E to get back to the trailhead.