Question 1203114
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Diagram
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Points<ul><li>A = the top of Ms. Cane's chimney</li><li>B = the top of Mr. John's chimney</li><li>C = Santa's location</li><li>D and E = needed to form right triangles ACD and BCE respectively. </li><li>F = point directly under point C, and the same horizontal level as A and B</li></ul>Angles<ul><li>angle ACD = 5.3 degrees (one of the angles of depression)</li><li>angle BCE = 10.2 degrees  (the other angle of depression)</li><li>angle CAF = 5.3 degrees</li><li>angle CBF = 10.2 degrees</li></ul>Segments<ul><li>AF = x</li><li>FB = 25-x</li><li>AB = 25 kilometers</li><li>CF = h = unknown height or altitude</li></ul>
<font color=red size=4>The goal is to calculate the value of h.</font>


An angle of depression is where you start looking at the horizon, then you move your viewpoint downward that number of degrees until reaching the target. 
This explains how angles ACD and BCE are set up.


Angle CAF is congruent to angle ACD because of the alternate interior angle theorem. AB is parallel to DE since both are horizontal.
Also, angle CBF = angle BCE for similar reasoning.


Focus on right triangle ACF
tan(angle) = opposite/adjacent
tan(angle CAF) = CF/AF
tan(5.3) = h/x
<font color=blue>h = x*tan(5.3)</font>
We'll use this in a <font color=blue>substitution</font> step later on.


Now focus on right triangle BCF
tan(angle) = opposite/adjacent
tan(angle CBF) = CF/FB
tan(10.2) = h/(25-x)
tan(10.2) = <font color=blue>x*tan(5.3)</font>/(25-x) <font color=blue>...... Substitution: replace h with x*tan(5.3)</font>
tan(10.2)*(25-x) = x*tan(5.3)
25*tan(10.2)-x*tan(10.2) = x*tan(5.3)
25*tan(10.2) = x*tan(5.3)+x*tan(10.2)
25*tan(10.2) = x*( tan(5.3)+tan(10.2) )
x = 25*tan(10.2)/( tan(5.3)+tan(10.2) )
x = 16.4953525945108
This value is approximate.
Make sure your calculator is in degree mode.


That x value leads to
h = x*tan(5.3)
h = 16.4953525945108*tan(5.3)
<font color=red>h = 1.5302275941042</font>
This value is approximate.


<font color=red>Santa's altitude or height is <font size=4>approximately 1.5302 km</font> </font>
Round this value however your teacher instructs.


Extra info
1.5302 km = 5020.34121 feet (approximate)
1.5302 km = 0.9508222 miles (approximate)
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