Please find the attached diagram for a better understanding of the question.
As we can see from the diagram,
RQ = 21 feet = height of the hill
PQ = 57 feet = Distance between you and the base of the hill
SR= h=height of the statue
[tex] \angle SPR [/tex]=Angle subtended by the statue to where you are standing.
[tex] \angle x=\angle RPQ [/tex]= which is unknown.
Let us begin solving now. The first step is to find the angle [tex] \angle x [/tex] which can be found by using the following trigonometric ratio in [tex] \Delta PQR [/tex] :
[tex] tan(x)=\frac{RQ}{PQ} =\frac{21}{57} [/tex]
Which gives [tex] \angle x [/tex] to be:
[tex] \angle x=tan^{-1}(\frac{21}{57})\approx20.22^{0} [/tex]
Now, we know that[tex] \angle x [/tex] and [tex] \angle SPR [/tex] can be added to give us the complete angle [tex] \angle SPQ [/tex] in the right triangle [tex] \Delta SPQ [/tex].
We can again use the tan trigonometric ratio in [tex] \Delta SPQ [/tex] to solve for the height of the statue, h.
This can be done as:
[tex] tan(\angle SPQ)=\frac{SQ}{PQ} [/tex]
[tex] tan(7.1^0+20.22^0)=\frac{SR+RQ}{PQ} [/tex]
[tex] tan(27.32^0)=\frac{h+21}{57} [/tex]
[tex] \therefore h+21=57tan(27.32^0) [/tex]
[tex] h\approx8.45 ft [/tex]
Thus, the height of the statue is approximately, 8.45 feet.
