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 1.00 miles. Next, you turn to your right by 160° and follow a 3rd trail for 2.20 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.

(a) Turn ?? to the right

(b)the walk ?? miles

Explanation / Answer

Let us consider the starting point as the origin of the coordinate system. Let positive x axis be towards right and positive y axis vertically upwards

Trail 1 - distance travelled in x direction =0

Distance travelled in y direction =2 miles

Trail 2 - distance travelled along x direction = -1*sin30 =-0.50 mile

Distance travelled along y direction =1*cos30=0.866 mile

Trail 3 - distance travelled along x direction = 2.2*cos40=1.685 miles

Distance travelled along y direction =-2.2sin40=-1.414miles

X coordinate after 3 trails =-0.5+1.685=1.185miles

Y coordinate after 3 trails =2+0.866-1.414=1.452miles

Distance to be travelled =sqrt(1.185^2 + 1.452^2)=1.874 miles

Angle by which he has to turn to right =180-40-arctan(1.452/1.185)=89.22 degreed

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