Question 1200155
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A compass bearing of something like S 52°E means we start facing directly south, then we turn 52 degrees toward the east.


Refer to this page for a few examples
<a href = "http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html">http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html</a>


This is one way to draw out the diagram.
*[illustration Screenshot_242a.png]
The diagram is to scale. It was made with GeoGebra.


The distance AB = 600 is due to the scratch work calculation shown below
distance = rate*time
distance = (250 mph)*(2.4 hrs)
distance = 600 miles


Angles EAC, DAB, and GBC are the compass bearings given to us. They are marked in blue shown above.
<table border = "1" cellpadding = "5">
<tr><td>Angle</td><td>Compass Bearing</td></tr>
<tr><td>EAC = 52</td><td>S 52°E</td></tr>
<tr><td>DAB = 84</td><td>N 84°E</td></tr>
<tr><td>GBC = 38</td><td>S 38°W</td></tr>
</table>The last two letters of each angle name tells us where we are located and where we are looking at in that exact order.
Example: Angle EAC has "A" and "C" as the last two letters. We're located at A and look toward C.


Vertical line DE is parallel to vertical line FG.
Segment AB is a transversal cut to these parallel lines.
Same side interior angles DAB and FBA are supplementary (because of the parallel lines).
So,
(angle DAB)+(angle FBA) = 180
(84)+(angle FBA) = 180
angle FBA = 180-84
angle FBA = 96



Angle BAC is found by noticing angles EAC, BAC, and DAB are supplementary


In other words,
(angle EAC)+(angle BAC)+(angle DAB) = 180
and similarly
(angle GBC)+(angle CBA)+(angle FBA) = 180
these two equations help determine angles BAC and CBA (44 and 46 respectively)


Then we use the idea that the inner angles of a triangle add to 180
A+B+C = 180
44+46+C = 180
90+C = 180
C = 90
Triangle ABC is a right triangle.



Since ABC is a right triangle, we have two options using the trig ratios sine and cosine
<table border = "1" cellpadding = "5">
<tr><td>Option 1</td><td>Option 2</td></tr>
<tr><td>sin(angle) = opposite/hypotenuse<br>sin(angle ABC) = AC/AB<br>sin(46) = x/600<br>x = 600*sin(46)<br>x = 431.60388020319<br>x = 431.6</td><td>cos(angle) = adjacent/hypotenuse<br>cos(angle BAC) = AC/AB<br>cos(44) = x/600<br>x = 600*cos(44)<br>x = 431.60388020319<br>x = 431.6</td></tr>
</table>Make sure your calculator is in degree mode.


Here's one possible route if you wanted to use the law of sines
sin(B)/b = sin(C)/c
sin(46)/x = sin(90)/600
600*sin(46) = x*sin(90)
x = 600*sin(46)/sin(90)
x = 431.60388020319
x = 431.6


Keep in mind that sin(90) = 1, so this calculation isn't entirely new compared to option 1 mentioned earlier.



Answer: <font color=red size=4>Approximately 431.6 miles</font>
Round this value however your teacher instructs.
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