From the top of a building "h1" = 21m tall, the angle of elevation of the top of a taller building is β = 46°. The angle of depression of the base of the taller building is α = 39°. what is the height "h" of the taller building?
Between the viewer on the building at height "h1" = 21m and the taller building are two imagined right triangles.
The ankathete "a" of the lower triangle (α = 39 °) is the distance "a" of the two buildings. The counter-cat "h1" = 21m is the height of the observer.
Then:
\({\color{blue}a=\frac{h1}{tan\ \alpha} = \frac {21m}{tan \ 39°}}\)
\({\color{blue}a=25.933\ m}\)
The upper part of the taller building "h2" is the counter-cathedral of the upper triangle. The distance "a" is the ankathete. β = 46 °.
Then:
\({\color{blue}h2=a\times tan\ \beta}\)
\({\color{blue}h2=25.933\ m\times tan\ 46°}\)
\({\color{blue}h2=26.854\ m}\)
\({\color{blue}h=h1+h2=21\ m+26.854\ m}\)
\({\color{blue}h=47.854\ m}\)
The height of the larger building is \({\color{blue}h=47.854\ m}\)
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