10th Grade Honors Geometry - Please help!?

Question

Two observers are 200 ft. apart, in line with a tree containing a bird's nest. The angles of elevation to th bird's nest are 30 degrees an 60 degrees. How far is each observer from the base of the tree?

Need Work to Show!
Solution is 100 ft. and 300 ft.

Answer 1

HONORS..... that's an important word....

If you draw the picture, with 60 degree angle on the left, 30 degree on the right, and the tree as a vertical line in between, you'll see that you have one big 30-60-90 triangle, and two smaller ones inside of it.

Let x be the distance from the guy on the left (60) to to the tree, and 200 - x is the distance from the guy on the right (30).

Since the hypotenuse of the big 30-60-90 triangle is 200, the shorter side (which is the line from the guy on the left to the top of the tree) must be 100, and the longer side would be 100√3.

So the 100 forms the hypotenuse of the 30-60-90 on the left, so the shorter side, which is x, must be 100/2 = 50 ft away.

Therefore the other guy must be 200-50 = 150 ft away

Answer 2

It may help for you to draw this out if you have trouble visualizing it.

So what we have is three points on a line O1, O2, and T.
We know that the distance from O1 to O2 is 200 ft.
Let's say that O2 to T = l.
We also have a bird's nest at height h.

tan(60) = opp/adj = h/l
l*tan(60)=h

we can see that sin(60) = h/sqrt(h^2+l^2)
we can also see that sin(30) = h/sqrt(h^2+(l+200)^2)

sin(30)*sin(60)/sin(30) = sin(60)

h/sqrt(h^2+l^2) = h/sqrt(h^2+(l+200)^2)*sin(60)/sin(30)
let's divide both sides by h and raise to the -1 power

sqrt(h^2+l^2) = sqrt(h^2+(l+200)^2) * sin(30)/sin(60)
square both sides

h^2+l^2 = (h^2+(l+200)^2) * (sin(30)/sin(60))^2
plug in l*tan(60) for h

solving the ugly equation that results is left as an exercise for the reader.

the distance for the closer observer = l and the further observer = l+200


EDIT: this was done assuming that the observers are on one side of the tree since that's what "in line with" means to me, but I could be wrong.

Answer 3

Maybe you will need to consider the two possible options --
1) The tree being between the two observers
2) Both observers being on same side of the tree

Other than that, you should be able to solve this using simple properties of right angle triangles

Answer 4

Make a sketch and notice that relationship between the tangents of the two angles, the height of the tree, and the distances along the ground.

Drawing a sketch for problems like this is almost always the best first step!