Question 1204918
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Diagram
{{{
drawing(400,400,-5,5,-5,5,

line(-3,-2,2,-2),line(2,-2,2,2),line(2,2,-1,-2),line(2,2,-3,-2),

line(1.6,-2,1.6,-1.6),line(1.6,-1.6,2,-1.6),

locate(-3.14,-2.28,"A"),locate(-1.2,-2.3,"B"),locate(1.96,-2.38,"C"),locate(2.2,2.36,"D"),locate(-2.2,-1.5+0.2,21^o),locate(-0.4,-1.54+0.2,43^o),locate(-2.2,-2.3+0.2,"4.5"),locate(0.38,-2.36+0.2,"x"),locate(2.28,0.28,"y"),

locate(-4,-4,matrix(1,4,"Diagram","not","to","scale"))

)


}}}

Josephine starts at point A. Then moves to point B.
C is the base of the mountain
D is the peak


The 43 degree angle is from computing 21+22 = 43.


Focus on right triangle BCD
tan(angle) = opposite/adjacent
tan(B) = CD/BC
tan(43) = y/x
y = x*tan(43)


Now focus on right triangle ACD.
tan(angle) = opposite/adjacent
tan(A) = CD/AC
tan(A) = CD/(AB+BC)
tan(21) = y/(4.5+x)
tan(21) = x*tan(43)/(4.5+x)
(4.5+x)*tan(21) = x*tan(43)
4.5*tan(21)+x*tan(21) = x*tan(43)
4.5*tan(21) = x*tan(43)-x*tan(21)
x*tan(43)-x*tan(21) = 4.5*tan(21)
x*(tan(43) - tan(21)) = 4.5*tan(21)
x = 4.5*tan(21)/(tan(43) - tan(21))
x = 3.148428 kilometers approximately


y = x*tan(43)
y = 3.148428*tan(43)
y = 2.935957 kilometers approximately


When rounding to 3 decimal places, we get 2.936 km which converts to 2936 meters. 
Not too far off compared to the actual height 2954 meters.
The discrepancy could be based on these factors<ul><li>Rounding error</li><li>Josephine might have made a mistake measuring either of the angles or the 4.5 km distance</li></ul>


The shortcut formula to find the height is
{{{height = d*tan(B)*tan(A)/(tan(B)-tan(A))}}}
d = distance between observation points
A and B = angles of elevation where B > A


More practice
<a href = "https://www.algebra.com/algebra/homework/Vectors/Vectors.faq.question.1198061.html">https://www.algebra.com/algebra/homework/Vectors/Vectors.faq.question.1198061.html</a>
and
<a href = "https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1198040.html">https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1198040.html</a>
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