SOLUTION: Dear Sir/madam; A pleasant day! please help me with this word problem about right triangle and angles of elevation. Here it goes: At the same time, Jerry and Lena, wh

Question 826185: Dear Sir/madam;
A pleasant day! please help me with this word problem about right triangle and angles of elevation. Here it goes:
At the same time, Jerry and Lena, who are 350 meters apart find the angles of elevation of a balloon between them at 35� and 46�, respectively. Find the height of the balloon if it is in the same vertical plane as them.
Thank you so much for your help!

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
You will probably understand this better if you draw a diagram:

On the same horizontal line, label two points, J and L (to represent Jerry and Lena).
Somewhere above the horizontal line and between points J and L, label a point B (for balloon).
On the horizontal line, directly below point B, label a point G (for ground?).
Draw segments to connect J and B, L and B, and B and G.
Label angle BJG as 35 degrees and angle BLG as 46 degrees.
Label the length of segment BG as "x".
Label the length of segment JG as "y"

Since we are asked to find the height of the balloon we will be trying to find the value of "x" in the diagram.

Our diagram has two right triangles: BJG and BLG. "x" is a side of both of these triangles. But we do not know any of the other sides of either of these triangles. So we are not yet ready to write an equation with just "x" and solve for it.

We have been told that the distance between Jerry and Lena is 350 meters. In the diagram, the 350 meters would be the length of segment JL. But JL is not a side of either right triangle. JL is, however, the sum of the length of JG and the length of GL. We have already labeled JG as "y". This makes GL: 350 - y. (Think about it.)

We now have two right triangles with known angles and with expressions in x and y for the opposite and adjacent sides of those angles. Since tan is opposite/adjacent, we can now write two equations, one for each triangle, for the tan of the known angles.
In triangle BJG:

In triangle BLG:

With two equations of two variables, we should be able to solve this. I will be using the Substitution Method to solve this system. And, since we are only really interested in what x is, I will solve for y first. (This will make sense in a minute.) Solving the first equation for y:

Multiply both sides by y:

Divide both sides by tan(35):

Now we'll substitute this into the second equation:

With only one variable left in the equation, x, we can now we solve for it. (Do you see now why I solved for y to make the substitution?) Multiplying both sides by the denominator of the right side:

Using the Distributive Property we get:

Multiplying both sides by tan(35) [to eliminate the remaining fraction:

Adding x*tan(46) [to get the x terms on the same side]:

Factor out x:

Dividing both sides by 1+tan(46):

This is an exact expression for the height of the balloon.

But we probably want this as a decimal approximation. So we'll reach for our calculators to find the tan's:
[tan's rounded to 4 places]
Now we simplify:

So the balloon is approximately 125 meters high.

P.S. If you wanted/needed to find y and/or 350-y then use the value we found for x and one of our two-variable equations.
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