# Find The Height Of The Mountain

1. Jul 22, 2012

### chikis

1. The problem statement, all variables and given/known data
A simple measuring device is used at point X and Y on the same horizontal level to measure the angles of elevation of the peak P of a certain mountain. If X is known to be 5,200m above sea level, |XY| = 4,000m and the measurement of the angles of elevation of P at X and Y are 15¤ and 35¤ respectively, find the height of the mountain. (take tan 15¤ = 0.3 and tan 35¤ = 0.7)

2. Relevant equations

3. The attempt at a solution

I drew a large triangle XPY which consist of two small triangles, XPO and POY.
The distance |XY| is 4000m. |XY| is above sea level of heigt 5,200m.
The height am required to calculate is the distance |OP|. Let |OP|= h.
We need to find the distance XO and OY respectively. Let XO = (4000-b) and OY = b. Considering triangle XPO, tan15 = h/(4000-b)
---> (4000-b)0.3 = h
1200-0.3b = h
b = 1200-h/0.3 ----->(1)
Considering triangle POY tan35¤ = h/b
h = btan35¤
---> h = 0.7b
b = h/0.7 ----->(2)
Equating equation 1 and 2, you will have:
1200-h/0.3 = h/0.7
840 - 0.7h = 0.3h
0.7h + 0.3h = 840m
h = 840m
the height of the mountian will now become 840m +5,200m
= 6040m
but my answer is not the same as the answer provided for the working. How do I aproach this problem?

2. Jul 22, 2012

### HallsofIvy

Staff Emeritus

3. Jul 23, 2012

### chikis

Yes O is the foot of the pependicular from P and meets the line XY at O.
But the questions says that X and Y are on the same horizontal level and that XY is 4000m, that makes the it difficult to understand between X and Y which is closer and which is farer.

4. Jul 23, 2012

### eumyang

See bolded. Because the angle of elevation at Y is larger, Y is closer to O than X.

5. Jul 23, 2012

### chikis

So if the angle of elevation at Y is larger and Y is closer to O than X, what do I make of that?

6. Jul 23, 2012

### chikis

7. Jul 23, 2012

### chikis

Code (Text):
P
/|\
/|   \
/  |     \
/   |       \
/    |         \
/__ _| _  _  _ _\
X     O           Y
That's how my diagram looks like.

Last edited by a moderator: Jul 23, 2012
8. Jul 23, 2012

### Staff: Mentor

X and Y should be on the same side of the origin, (and the same side of the mountain). The diagram that you drew would not be useful to surveyors.

Point Y is closer to the mountain than point X, since the angle of elevation at Y is larger than the angle of elevation at X.

The idea is that the surveyors set up their equipment at point Y, determine the angle to the top of the mountain, and then move farther away. At point X, they get the new angle of elevation (which is smaller).

Last edited: Jul 24, 2012
9. Jul 23, 2012

### eumyang

chikis, see attached diagram.
(Mods, I hope this is okay. He/she's tried to draw a diagram for us.)
I think you mean point X.

#### Attached Files:

• ###### Surveyor.png
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10. Jul 23, 2012

### Staff: Mentor

IMO, it's fine.
Right you are - I meant X.

Edit: I've gone back and fixed that error.

Last edited: Jul 24, 2012
11. Jul 24, 2012

### chikis

But the question never said that X and Y are on the same side of the mountain. The question says that the angle of elevation of P at X is 15¤ and that the angle of elevation of P at Y is 35¤ and XY is 4000m which fits my diagram excatly.
With my diagram, if I use tan15 = h/(4000+b) tan35 = h/b, I will get 7300m as my final answer, but if use tan15 = h/(4000-b) and tan35 = h/b, I will get 6040m as the final answer. I think am actually confuse on how to get the distance XO and OY, is there any help I can get on that.

12. Jul 24, 2012

### eumyang

Read mark44's post (#8) again. If X and Y were to be on opposite sides of O, the problem would have mentioned that.

With your diagram, that's impossible. You would end up with |XO| being longer than |XY|.

You said it yourself in the original post!
BTW, is 7,300m the answer that the book provided?

13. Jul 24, 2012

### chikis

No, the book provided 6400 as the final answer which I think is not correct.

14. Jul 24, 2012

### eumyang

It isn't. I'm getting 7,300 m as well (using tan 15° ≈ 0.3 and tan 35° ≈ 0.7). Must be a typo.

15. Jul 24, 2012

### chikis

You must note that you will get 7,300m if only and only if X and Y are on same side of the mountain but if they at the opposite sides of the mountain as stated clearly in the question, then you will get 6040m which ever way you compute XO and OY. If you like take XO as tan15¤ = h/4000-b and tan35¤ = h/b or if you like take tan35¤ = h/4000-b and tan15¤ = h/b, you will still get 6040m.

16. Jul 24, 2012

### eumyang

Really? I'm looking at the original post...
... and I don't see it stated clearly that X and Y are on opposite sides of the mountain.

Once more I ask you, did you read Mark44's post (#8, emphasis mine)?

Last edited by a moderator: Jul 24, 2012
17. Jul 24, 2012

### chikis

Yes I did, and let me ask as well, is Mark44 a surveyor or an engineer to know that surveyors set up their equipment at point Y, determine the angle to the top of the mountain, and then move farther away. At point Y, they get the new angle of elevation (which is smaller).
Assuming X and Y are at the opposite sides of the mountain and XY is 4000m, how do you compute XO and OY respectively?

18. Jul 24, 2012

### Ray Vickson

It does not matter whether Mark44 is an surveyor, or what. By lots of experience, he can say that if the points X and Y are taken on opposite sides of the mountain, the question would have said so (or else the person setting the question was sloppy). In fact, the question leaves open the positions of X and Y; P could be due North of X and Y could be NNE of X, but closer to P, so there are probably infinitely many "solutions" that fit the original description (depending on the angle XOY). Basically, it is an unstated assumption that we want the simplest possibility that would also be physically the simplest: measuring the distance |XY| by surveying requires X and Y to be on the same side of the mountain; otherwise, some much more involved operations would be needed to measure |XY| when X and Y are on opposite sides of the mountain and thus invisible to each other. Furthermore, the mathematically simplest situation is when X, Y and O are on the same straight line.

RGV

19. Jul 24, 2012

### chikis

Ok! Since the question did not state clearly the position of X and Y. We take it that they are two possible solutions, that for when X and Y are on same sides of the mountain, we have known, but what of when X and Y are at opposite sides of the mountain?

20. Jul 24, 2012

### Ray Vickson

Then you have a triangle XYP with interior angles of 15 degrees and 35 degrees given at X and Y, and you are given the length of the base XY. If h is the height (above the base) and |XO| = x, we have h/x = tan(15) and h/(4000-x) = tan(35). You can solve these two equations to get h and x.

RGV