# Thread: height of a mountain

1. ## height of a mountain

From a point on ground level, you measure the angle of elevation to the top of a mountain to be 35º. Then you walk 160 m farther away from the mountain and find that the angle of elevation is now 15º. Find the height of the mountain.

I keep getting 30, but it is not correct. I can't figure out where the 15 degrees goes. Do I have it drawn correctly thus far? Thanks much.

2. Hello, daydreembelievr!

Your diagram is wrong . . .

From a point on ground level, the angle of elevation to the top of a mountain is 35°.
160 m farther away from the mountain, the angle of elevation is 15º.
Find the height of the mountain.
Code:
    P *
| *  *
|   *     *
h|     *        *
|       *           *
|     35° *         15°  *
* - - - - - * - - - - - - - - *
Q     x     A       160       B

The height of the mountain is: $h = PQ.$

The first observation is made at $A\!:\;\;x = QA,\;\angle PAQ = 35^o$

The second observation is made at $B\!:\;\;AB = 160,\;\angle PBQ = 15^o$

In right triangle $PQB\!:\;\;\tan15^o \:=\:\frac{h}{x+160} \quad\Rightarrow\quad x \:=\:\frac{h}{\tan15^o} - 160$ .[1]

In right triangle $PQA\!:\;\;\tan35^o \:=\:\frac{h}{x}\quad\Rightarrow\quad x \:=\:\frac{h}{\tan35^o}$ .[2]

Equate [1] and [2]: . $\frac{h}{\tan15^o}-160 \:=\:\frac{h}{\tan35^o} \quad\hdots\quad \text{and solve for }h$

3. ## unsure

Sooo helpful. I forgot to make it x+160 instead of just x. But now when I try to solve it I keep getting 25.9 which is close to what I was getting before...but I know that isn't the answer (I have the answer key). Am I doing something wrong?

h/0.7002 = h/0.26795 - 160

h=(0.7002)h/0.26795 - 160
h= 0.7002h/0.26795 - 112.032
0.26795h=0.7002h-112.032
h-25.9

thanks sooo much.

4. Hello, daydreembelievr!

We have: . $\frac{h}{\tan15^o}-160 \:=\:\frac{h}{\tan35^o}$

Multiply by $\tan15^o\tan35^o\!:\quad h\tan35^o - 160\tan15^o\tan35^o \:=\:h\tan15^o$

. . . . . . $h\tan35^o - h\tan15^o \:=\:160\tan15^o\tan35^o$

Factor: . $h(\tan35^o - \tan15^o) \:=\:160\tan15^o\tan35^o \quad\Rightarrow\quad h \:=\:\frac{160\tan15^o\tan35^o}{\tan35^o-\tan15^o}$

Therefore: . $h \;=\;69.44737423...$