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 1.40 miles. Next, you turn to your right by 160° and follow a 3rd trail for 1.80 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.

Turn ____ degrees to the right and then walk ____ miles. Thank you in advance!

Explanation / Answer

In this case, you turned left 30 degrees from your original path and turning right 160 degrees. It implies that your line of travel is 160 - 30 = 130 degrees from your original path.

If you need to find number of degrees from right 360 - 130 = 230 degree

At turn about point your distance traveled parallel to your original path is as follow

d1 = 2.00 + 1.40 cos30 + 2.00 cos(-130)

d1 = 2.569 miles

Distance traveled perpendicular to the original path:

d2 = 0 + 1.40 sin30 + 2.00sin(-130)

d2 = - 0.832 miles

Therefore, your distance back to the starting point is

d = (2.569 miles)² + (- 0.832 miles )²)

d = 2.70 miles

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