Question

cheyenne and jacob are geocaching. they are currently hiking on a main trail at a bearing of n 45° e. they determine that the cache should be about 1.5 miles off the trail at a bearing of n 17° e. if they want to hike to a point 3 miles further along the main trail once they have located the cache, they will hike an additional. geo - caching is a sport that involves hiking and using clues to find hidden “treasures.” the person who finds the cache is then obliged to leave a treasure for the next person. (there is a diagram with geo - cache marked by a star, a 1.5mi line from the start point to geo - cache, a 3mi line labeled main trail from the start point along a green line, and the diagram is not to scale.)

Explanation:

Step1: Find the angle between the two paths

The bearing of the main trail is \( N 45^\circ E \) and the bearing to the cache is \( N 17^\circ E \). The angle between them is \( 45^\circ - 17^\circ= 28^\circ \).

Step2: Apply the Law of Cosines

We have a triangle with sides \( a = 3 \) miles (along the main trail), \( b = 1.5 \) miles (to the cache), and we need to find the side \( c \) (the additional hike). The Law of Cosines is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), where \( C = 28^\circ \), \( a = 3 \), \( b = 1.5 \).
First, calculate \( \cos(28^\circ)\approx0.8829 \)
Then, \( a^{2}=3^{2} = 9 \), \( b^{2}=1.5^{2}=2.25 \), \( 2ab\cos(C)=2\times3\times1.5\times0.8829 = 8.0\times0.8829=7.9461 \)
So \( c^{2}=9 + 2.25-7.9461=3.3039 \)

Step3: Find the length of \( c \)

Take the square root of \( c^{2} \): \( c=\sqrt{3.3039}\approx1.82 \) miles.

Answer:

\( \approx1.82 \) miles