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 degree to follow a 2nd trail for 1.00 miles. Next, you turn to your right by 160 degree and follow a 3rd trail for 1.70 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 to the right, then walk

Explanation / Answer

Let us say the first path points in the direction of the +x-axis.

End of first path: Ax = 2.00 mi

End of second path:

Ax = 2.00 + 1.00 cos30.0° mi

= 2.866 mi

Ay = 1.00 sin30.0° mi = 0.50 mi

End of third path:

Ax = 2.866 mi + 1.70 cos (30.0° - 160°) mi

= 1.77 mi

Ay = 0.50 mi + 1.70 sin(30.0° - 160°) mi

= -0.80 mi

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Now find the reference angle to assist you get back to where you started:

Tan = Ay / Ax

= (-0.80 mi) / (1.77 mi)

= -24.3°

    = 24.3° clockwise from +x-axis

However, we have to go the other direction: 180° - 24.3° = +155.6°

and you are facing the direction, 30.0° - 160° = -130°

(Or) 360° - 130° = +230° at the end of the third path, so you have to turn the direction at angle

230° - 155.6° = 74.4° to the right

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The distance walked can be calculated from the coordinates after end of the third path:

A = sqrt [(Ax)² + (Ay)²]

= sqrt [(1.77 mi)² + (-0.80 mi)²]

= 1.94 mi

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