Answer: To solve this problem, we can use the law of cosines. The law of cosines is a useful tool for solving problems involving distances and angles between points in two-dimensional space. We can use the law of cosines to find the distance between the starting and ending point of the hike, and the bearing to get back to camp.
The law of cosines states that:
c^2 = a^2 + b^2 - 2ab * cos C
where c is the distance between the two points, a and b are the distances of the two legs of the hike, and C is the angle between them.
Using the given information, we can set a as the distance traveled at 3.5 miles per hour for 2 hours, which is 3.5 * 2 = 7 miles,
b as the distance traveled at 3.5 miles per hour for 4 hours which is 3.5 * 4 = 14 miles,
C as the angle between the two legs of the hike which is (72 + 23) = 95 degrees.
Now we can substitute this values into the law of cosines formula:
c^2 = 7^2 + 14^2 - 2(7)(14) * cos 95
Solving this equation will give us the square of the distance between the starting and ending point of the hike. To find the actual distance, we'll have to take the square root of that.
The distance directly back to camp is √(196 - 98*(√2/2)) = √(196 - 98*(√2/2)) ≈ 8.14 miles
To find the bearing to head back to camp, we must know that the bearing of a line is defined as the angle between the line and the North direction, measured clockwise.
We can use the bearing of N72°W and S23°W to find the back bearing to camp.
The back bearing is (180 + 72 + 23) = 275 degrees. The hikers should take a bearing of 275 degrees to head back to camp.
Explanation: