Question

In the sport of orienteering, participants must plan carefully to get from one checkpoint to another in the shortest possible time. In one case, starting at a particular checkpoint, Sam decides to take a straight path that goes west for 690 meters, and then go northeast for 590 meters on another straight path to reach the next checkpoint. Northeast, by the way, means 45 degrees to both north and east. Between the same two checkpoints, Mary decides to take the shortest distance between the two checkpoints, traveling off the paths through the woods instead. (a) What is the distance that Mary travels between the checkpoints? (Note that Sam travels a total distance of 1280 m, although his speed is probably significantly higher than Mary's.) m (b) In what direction does Mary go? Specify her direction as a positive angle, measured counterclockwise from east.

Answer 1

Let $\vec{A}$ represent Sam's first path

$\vec{A}=690m\ west$

${\color{Red} \vec{A}=-690i+0j}$

and let $\vec{B}$ represent Sam's second path

$\vec{B}=590m\ northeast$

$\vec{B}=590m\ at\ 45^{\circ}\ from\ East\ axis$

$\vec{B}=B_{x}i+B_{y}j=Bcos\theta i+Bsin\theta j=590cos45i+590sin45j$

${\color{Red} \vec{B}=417.193i+417.193j}$

==================

Resultant of these two vectors is the straight path taken by Mary

$\vec{R}=\vec{A}+\vec{B}$

$\vec{R}=(-690i+0j)+(417.193i+417.193j)$

$\vec{R}=-690i+0j+417.193i+417.193j$

$\vec{R}=(-690+417.193)i+(0+417.193)j$

${\color{Red} \vec{R}=-272.807i+417.193j}$

----------------

Magnitude

$R=\sqrt{R_{x}^{2}+R_{y}^{2}}$

$R=\sqrt{(-282.807)^{2}+(417.193)^{2}}$

$R=306.71m$

(a) ANSWER :${\color{Red} 306.71m}$

===============

Direction

$\theta =tan^{-1}(\frac{R_{y}}{R_{x}})$

$\theta =tan^{-1}(\frac{417.193}{-282.807})$

$\theta =-55.87^{\circ}$

$\theta =180^{\circ}-55.87^{\circ}$

ANSWER: ${\color{Red} \theta =124.13^{\circ}\ from\ East \ axis}$

==========================================