Question

A tree on a hillside casts a shadow c = 230 ft down the hill. If the angle of Inclination of the hillside is b = 20° to the horizontal and the angle of elevation of the sun is a = 50°, find the height of the tree. (Round your answer to the nearest foot.)

Answer 1

179 ft

Step-by-step 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:

`\begin{gathered} \cos \text{ 20 = }\frac{\text{adj}}{hyp} \\ \Rightarrow\text{ cos 20 = }(H)/(230) \\ =>\text{ H = 230 }\cdot\text{ cos 20} \\ H\text{ = 216.13 ft} \end{gathered}`

Now, we need to find G, because the height of the tree is:

height of tree = T - G

From the smaller diagram, using SOHCAHTOA:

`\begin{gathered} \sin \text{ 20 = }(opp)/(hyp)\text{ = }(G)/(230) \\ G=\text{ 230 }\cdot\text{ sin 20} \\ G\text{ = }78.66\text{ ft} \end{gathered}`

So, now, we use the big triangle to find T by using SOHCAHTOA:

`\begin{gathered} \tan \text{ 50 = }(opp)/(adj)\text{ = }(T)/(H) \\ \tan \text{ 50 = }(T)/(216.13) \\ T\text{ = 216.13 }\cdot\text{ tan50} \\ T\text{ = 257.57 ft} \end{gathered}`

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.