TRIG WORD PROBLEM HELp?

Question

I am in 8th grade and we are doing trig. its all pretty easy except im stuck on one problem:

Jane needs to know the height of a tree in her yard. From a point on the ground she finds the angle of elevation to the top of the tree is 36.7 degrees. She can move back 50 ft and from this second point the angle of elevation to the top of the tree is 22.2 degrees. Find the height of the tree. (Note: For this problem you do not have to account for Jane's height).

Please explain every step and even if you think i might know it mabey i dont and i really need this problem.

Answer 1

The tree itself, the ground between Jane and the tree, and Jane's line of sight to the top of the tree, are three sides of a right triangle.

You have two tangent equations, involving the height of the tree, the distance to the tree, and the angle that Jane measured:

tan(36.7) = height of tree / initial distance to tree

tan(22.2) = height of tree / (50 + initial distance to tree)

Let's call the height of the tree "h" and the initial distance "d". Then the two equations become:

tan(36.7) = h / d
tan(22.2) = h / (50 + d)

And calculating the tangent values, they are (to four decimal places):

0.7454 d = h
0.4081 (50 + d) = h

Since both equations are in terms of h, we can say that:

0.7454 d = 0.4081 (50 + d)
(0.7454 - 0.4081) d = (0.4081) 50
(0.3373) d = 20.405
d = 60.5

Now that we know d, we can plug it into either equation to find h:

0.7454 d = h
0.7454 (60.5) = h
45.1 = h

The tree is about 45 feet tall.

Answer 2

It would definitely help you (as usual in trig) to draw a picture. If you do this, you will see we have two right triangles. Then you can write out the equations for the tangents of each triangle. You will see that

tan(36.7deg) = h/x
and
tan(22.2deg) = h/(x+50),
where h is the height of the tree in feet, and x is Jane's distance from the tree when she sees the 36.7 degree angle.

Now you use a calculator and find the tangent values. This gives you

h = (x+50)tan(22.2) = xtan(36.7)

Now you can easily solve for x, and when you do you can plug it back into one of your two equations to find h.

Answer 3

tan60=y/250 250tan60=y y=80 y+5.5=~86