SOLUTION: When Josephine stood south of the base of Mount Apo, in the Philippines, and looked up to the top, the angle of elevation was 21°. When Josephine drove 4.5 km closer, she found th

Question 1204918: When Josephine stood south of the base of Mount Apo, in the Philippines, and looked up to the top, the angle of elevation was 21°. When Josephine drove 4.5 km closer, she found that the angle of elevation increased by 22°. What is the height of Mount Apo? (Note: the actual height of Mount Apo is 2954m)
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39627) (Show Source): You can put this solution on YOUR website!
This example is very much like another solved a few days ago.

Study this one: https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1204848.html
https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1204848.html
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Diagram

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
Rounding error
Josephine might have made a mistake measuring either of the angles or the 4.5 km distance

The shortcut formula to find the height is

d = distance between observation points
A and B = angles of elevation where B > A

More practice
https://www.algebra.com/algebra/homework/Vectors/Vectors.faq.question.1198061.html
and
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1198040.html

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