Question

givings direccións to find different ranger staton.unluckily all the ranger are busy rescuing others campers and can't come to your aid so you will have to follow directions to find ranger station. The first response instructor you to travel north (90degrees) for 4.5 miles then travel 8.1mike at a direction of 125 degree. The second reponse tell you to travel 7.5 mile at a bearing of 20 degrees then 5.3miles at a bearing of 100degrees. Which ranger station is closest to your starting position (find the magnitude of the resultant vector of your displacement for each path ) explain each estro of your process,decide which path you will take ,and defend you choice

Answer 1

To find the resultant vector you need to add the horizontal and vertical components of each individual vector.
Cosine for horizontal, Sine for vertical.
$v = \ \textless \ |v|cos \theta, |v| sin \theta\ \textgreater$
Where |v| is magnitude or length of vector.
Lets look at first ranger station. First vector has magnitude 4.5 with angle of 90.
Next has magnitude of 8.1 with angle of 125.
$F_1 = \ \textless \ 4.5 cos(90)+8.1 cos(125) , 4.5 sin(90) + 8.1 sin(125) \ \textgreater$
$F_1 = \ \textless \ -4.646, 11.135\ \textgreater$
Find magnitude:
$|F_1| = \sqrt{((-4.646)^2+(11.135)^2} = 12.065$
Do the same with 2nd ranger station:
$F_2 = \ \textless \ 7.5 cos(20) + 5.3 cos(100), 7.5 sin(20) + 5.3 sin(100)\ \textgreater$
$F_2 = \ \textless \ 6.127, 7.784\ \textgreater$
$|F_2| = \sqrt{(6.127)^2 + (7.784)^2} = 9.9$

Therefore the 2nd ranger station is closest to starting point.