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° to follow a 2nd trail for 1.20 miles. Next, you turn to your right by 160° and follow a 3rd trail for 2.00 miles. At this point you are getting very tired and would like to get back as quickly as possible, but all of the available trails seem to lead you deeper into the woods. You would like to take a shortcut directly through the woods (ignoring the trails). How far to your right should you turn, and how far do you have to walk, to go directly back to your starting point? Feel free to use the provided vector drawing board to help visualize your work.

Explanation / Answer

As you have already turned left 30 degrees from your original path,

turning right 160 degrees means your line of travel is 160 - 30 = 130 degrees from your original path.

at turn about point your distance traveled parallel to your original path is

d1 = 2.00 + 1.00cos30 + 2.00cos(-130)

d1 = 1.58 miles

the distance traveled perpendicular to the original path is

d2 = 0 + 1.20sin30 + 2.00sin(-130)

d2 = -.93 miles

so your distance back to the starting point is

d = (1.58)² + (-0.93)²)

d = 1.83 miles

If we assume the original direction was zero degrees with positive angles in the counter clockwise direction,

your first turn had you heading +30°. Your second heading had you heading -130° which is the same as

360 - 130 = 230°.

Your new heading needs to be

tan = dy/dx

tan =0.93 / -1.58

=30.48°

which means we need to turn right

230° - 30.48° =199.52°

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