Trig word problem

KublaKhan

1. The problem statement, all variables and given/known data

A woman measures the angle of elevation of a mountaintop as 12.0°. After walking 0.80 km closer to the mountain on level ground, she finds the angle to be 14.0°
(a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles.

(b) Select variable names for the mountain height (suggestion: y) and the woman's original distance from the mountain (suggestion: x) and label the picture. (Do this on paper. Your instructor may ask you to turn in this work.)

(c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two selected variables. (Do this on paper. Your instructor may ask you to turn in this work.)

(d) Find the height y of the mountain by first solving one equation for x and substituting the result into the other equation.

3. The attempt at a solution

My two beginning equations are:
1. tan 12 = y/.8
2. tan 14 = y/x

The solution to equation 1 is .17
It seems to me that I no longer need to use two tangent equations as the problem suggests. I now know enough information for the following equation.

3. sin 14 = y/.17
Solution = .04km for the mountain height

I am way off the answer (which I know to be 1.15km)
What is wrong with my approach to the problem?
How do I substitute one equation into the other if I approach the problem in the suggested fashion?

KublaKhan

Holy Mackerel. I shouldn't do these things tired.

My first equation should be tan 14= y/x
2nd should be tan 12=y/(x-.8)
And my approach was obv wrong as I was trying to make a right triangle out of one that is not.
Still lost though. help?

As far as solving for x the farthest I can get is:
x tan 14=y
x = y / tan 14

I then would plug this into the other which just results in a wacky equation:

tan 12 = y / (y/tan 14) - .8

Not sure where to go from here

HallsofIvy

No. Since tangent is "opposite/adjacent", you want height of mountain over distance to the mountain. Height of the mountain is y but the distance to the mountain is NOT .8 but .8 closer than originally: x- .8.

Unfortunately, equation 1 is incorrect.

DecayProduct

I didn't get your known answer of 1.15 km, but I got pretty close. So, draw the mountain. The height of the mountain (y) forms the vertical side of a right triangle, the ground forms the horizontal. At some point, call it x, you measure an angle of 12 degrees. The height of the mountain, the distance x from the mountain, and the line from x to the peak form a right triangle. You walk 800 meters closer to the mountain, this new point, call it (x-800), measures 14 degrees to the peak. Now you have two right triangles that have the same y.

You also have two simultaneous equations: tan12 = y/x, and tan14 = y/(x-800). You should be able to rearrange these to get the result you are looking for.