Calculating the Height of a Leaning Tree using Trigonometry - Geometry Help

[MayQueen]

Geometry HELP! PLEASE!

Homework Statement

A tree is leaning 14 degrees from the vertical. The tree casts a 40 foot shadow on the flat ground. The line from the trees shadow to the top of tree creates a 60 degree angle above the horizontal. I need to find the length of the tree.

Homework Equations

The Attempt at a Solution

I tried everything, i know of but I am stuck. I tried using trig but that only gave the lenghts for the outside of the right triangle the tree divides. I need help desperately!

 

[Pi-Bond]

The question seems to be missing...you might also want to draw a diagram.

 

[MayQueen]

Ok , I am sorry I didnt realize , that i was the only one with the picture in front of me :) Here you go.

 

[Pi-Bond]

Do you know the sine rule?

 

[MayQueen]

No, can you explain to to me?

 

[Pi-Bond]

If you have a textbook, you should probably read through that. The link expresses the rule concisely:

http://www.ucl.ac.uk/Mathematics/geomath/trignb/trig11.html

 

[MayQueen]

So to use the sine rule i don't need a right triangle?

 

[Pi-Bond]

No, it can be applied to any triangle.

 

[MayQueen]

Im sorry that above question is rather stupid. However based on that explanation from the link, I don't understand this part :

That gives c=2sin(105o)/sin(30o)
which is 4sin(105o).

We can write sin(105o) as sin(150o-45o) then use the sin(A-B) rule to write this as
sin(150o)cos(45o)-cos(150o)sin(45o)
Im confused as to why we can write the above statement. Does it change the answer?

 

[Pi-Bond]

The two answers you get are equivalent, since you applied the identity to sin(105)

4sin(105o)=4( sin(150o)cos(45o)-cos(150o)sin(45o) )

 

[MayQueen]

Ok Great! then according to my calculations the length of the tree is 49.8677 ft

 

[PeterO]

Ok Great! then according to my calculations the length of the tree is 49.8677 ft

If when you drew your diagram, you made the 60 degree angle something like 60 degrees, rather than only 30 - 35 degrees, you would get some feedback from the diagram. [Notice that the angle you have marked as 44 appears BIGGER than the angle you have marked 60; it is even bigger than the angle you have marked 136]

Since the shadow is 40' long, if your calculation is correct the "tree" should appear longer. [in a perfect scale diagram, the tree would be close to 125% the length of the shadow]

If you are going to draw a diagram, at least make it close to the real thing - just so you get a pictorial indication that your calculation is correct.

Peter