One day you go hiking at a nearby nature preserve. At Tirst, you follow the stra

Question

One 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 30.0 to follow a 2nd for miles. 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 Number o to the right, Turn Number then walk miles 0.25 miles Note: The vector drawing board is meant as a uide and a reality check, but may not give you the answers with sufficient precision. It is preferable to calculate the magnitude and direction by hand.

Explanation / Answer

here,

let the upward be + y axis and the right be + x axis

the displacements are , d1 = - 2 mi j

d2 = 1.3 (cos(30) i + sin(30) j)

d2 = (1.13 i + 0.65 j) mi

d3 = 1.8 (cos(360 -160) i + sin(360 - 160) j)

d3 = -1.69 i mi - 0.62 j mi

the resultant of these vector is the shortest distance , d = - 0.56 i mi - 1.97 j mi

|d| = sqrt(0.56^2 + 1.97^2)

|d| = 2.05 miles

theta = arctan(1.97/0.56)

theta = 74.13 degree

he had to turn right at an angle (180 - theta)

then he had to turn 105.87 degree to the right

then he had to walk 2.05 miles

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