e day you go hiking at a nearby nature preserve. At Tirst, you follow the straig
ID: 1590670 • Letter: E
Question
e day you go hiking at a nearby nature preserve. At Tirst, 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 30.09 to follow a 2nd trail for 1.00 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 ead 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 Number Turn to the right Number then walk miles 0.25 miles Note: The vector drawing board is meant as a guide and a reality check, but may not give you the answers with sufficient precision. It is preferable to calculate the magnitude and direction by handExplanation / Answer
Keep track of position as you move along. Let's 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 + 1.00cos30.0° mi
Ax = 2.86 mi
Ay = 1.00sin30.0° mi
Ay = 0.5 mi
End of third path:
Ax = 2.86 + 2.00cos(30.0° - 160°) mi
Ax = 1.57 mi
Ay = 0.5 + 2.00sin(30.0° - 160°) mi
Ay = -1.032 mi
Reference angle to to help you get back to where you started:
tan = Ay / Ax
tan = (-1.032) / (1.57)
= -33.3° (33.3° clockwise from +x-axis)
But we need to go the other direction:
180 - 33.3° = +146.7°
And you're facing 30.0° - 160° = -130° (or 360° - 130° = +230°) at the end of the third path, so you'll need to turn
230° - 146.7° = 83.3° to the right
And walk a distance:
A = sqrt[(Ax)² + (Ay)²]
A = sqrt[(1.57 mi)² + (-1.032 mi)²]
A = 1.87 mi
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