one day you go hiking at a nearby nature preserve. at first, you follow the stra

Question

one day you go hiking at a nearby nature preserve. at first, you follow the straight, clearly marked trails. from your starting point, you travel 2.00 miles down the 1st trail. then you turn to your left by 30.0 degrees to follow a 2nd trail for 1.40 miles. next, you turn to your right by 160 degrees and follow a third trail for 1.90 miles. at this point you are getting very tired and would like to go back.

you would like to take a shortcut directly through the woods. how far to your right should you turn, and how far do you have to walk, to go directly back to the darting point.

what degree turn to the right must "I" take, and how many miles must I walk back?

I know the distance is 2.104 miles, but I cannot figure out the degree angle.

Explanation / Answer

Considering north and east as positive,
And taking the initial direction of the person as east,

net displacement in the eastern direction is

2 + 1.5 cos 30 + 1.8 cos (360 -160 + 30) = 2.14 m toward east.

Net displacement in the northern direction is
1.5 sin 30 + 1.8 sin (360 -160 + 30) = - 0.62880
Minus indicates that he has moved to south

Resultant ? (2.14 ^2 + [- 0.62880 ^2]) = 2.23
2.23 cos ? = 2.14
? = 16.3

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