Using the concept of height and distance and trigonometric functions, the height of the tree could be found. The height of the tree is 29 feet.
Given:
Height of observation point (eyes of student) is [tex]h=5[/tex] feet.
Distance between the student and the tree is [tex]d=20[/tex] feet.
The angle of elevation of the top of the tree is [tex]\theta= 50^{\circ}[/tex].
[tex]cos\theta=cos50^{\circ}=0.64[/tex]
Let the height of the tree be H.
See the figure attached.
The value of tangent of theta will be,
[tex]tan\theta=\dfrac{\sqrt{1-cos^2\theta}}{cos\theta}\\=\dfrac{\sqrt{1-0.64^2}}{0.64}\\=1.2[/tex]
Now, in triangle ABE, (see figure)
[tex]tan\theta=1.2\\\dfrac{AE}{d}=1.2\\\dfrac{AE}{20}=1.2\\AE=24[/tex]
So, the height H of the tree will be,
[tex]H=AE+h\\H=24+5\\H=29[/tex]
Therefore, the height of the tree is 29 feet.
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