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12 pts? Harris and Annabelle are 200 feet apart, and they are standing on opposite sides of a tree. The angles of elevation from Harris to the top of the tree is 30°, and the angle of elevation from Annabelle to the top of the tree is 35°. Find the height of the tree. Round your answer to the nearest tenth.

Respuesta :

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If we are to connect the root of the tree to a person to the top of the tree, we form two right triangles. If we let x and and 200 - x be the distance of Harris and Annabelle from the tree, respectively and h be the height, we form the trigonometric functions,
                                       tan 30° = h/x
                                       tan 35° = h/(200 - x)
The values of h and x in the equations are 63.28 ft and 109.62 ft, respectively. Thus, the answer to this item is approximately 63.28 ft. 

Answer: 63.3

Step-by-step explanation:

First, draw a triangle with the vertices at Harris, Annabelle, and the top of the tree. Notice, the angles are 30 degrees and 35 degrees. Therefore, the missing angle is 180 - 30 - 35 = 115 degrees

If we use the law of Sines, we can determine the distance from Harris to the top of the tree, x:

X/sin(35 degrees) = 200/sin (115 degrees)

x= 200sin(35 degrees)/ sin(115 degrees)= 126.57

Now, to find the height of the tree, we can drop the altitude of the triangle and just look at the smaller triangle created with the vertices at Harris, the top of the tree, and the base of the tree. Since this is a 30 - 60 - 90 triangle with x=. 126.57 ft as the length of the hypotenuse, then the height is half of that, 63.29 ft, since it is the short leg (across from the 30 degree angle).

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