ANSWER
179 ft
EXPLANATION
Let us draw a simplified diagram of the problem:
We first have to find H.
Let us split the triangle into two, to make it simpler:
To find H from the smaller triangle, use trigonometric functions SOHCAHTOA.
We have that:
[tex]\begin{gathered} \cos \text{ 20 = }\frac{\text{adj}}{hyp} \\ \Rightarrow\text{ cos 20 = }\frac{H}{230} \\ =>\text{ H = 230 }\cdot\text{ cos 20} \\ H\text{ = 216.13 ft} \end{gathered}[/tex]
Now, we need to find G, because the height of the tree is:
height of tree = T - G
From the smaller diagram, using SOHCAHTOA:
[tex]\begin{gathered} \sin \text{ 20 = }\frac{opp}{hyp}\text{ = }\frac{G}{230} \\ G=\text{ 230 }\cdot\text{ sin 20} \\ G\text{ = }78.66\text{ ft} \end{gathered}[/tex]
So, now, we use the big triangle to find T by using SOHCAHTOA:
[tex]\begin{gathered} \tan \text{ 50 = }\frac{opp}{adj}\text{ = }\frac{T}{H} \\ \tan \text{ 50 = }\frac{T}{216.13} \\ T\text{ = 216.13 }\cdot\text{ tan50} \\ T\text{ = 257.57 ft} \end{gathered}[/tex]
Therefore, the height of the tree is:
height = 257.57 - 78.66
height = 178.91 ft
To the nearest foot, the height of the tree is 179 ft.
