# Finding the lower and upper limits for angle of elevation

• songoku
The question tells you exactly what the angle is at 1,500m and at 1,800m but in order to answer the question you have to make an assumption about the upper and lower bounds on the angle at, for instance, 1,650m. In the absence of any other information there is only one sensible assumption you can make, can you see what that is?Assuming that the angle at 1,650m is the angle between the tangent line of that location and the path (i.e. the line connecting the starting point and the angle of elevation at 1,650m), the lower bound would be 12o and the upper bound will be 15o.f

#### songoku

Homework Statement
The base of a mountain starts at elevation of 900 metres above sea level and the top is at 15000 metres. The table below gives the angle of elevation of a path with respect to horizontal at different locations along a section of the route. The distances are measured along the path, starting from elevation of 1000 metres above sea level.

Find the lower and upper limit of angle of elevation at the end of the section
Relevant Equations
Riemann Sum

Integration
 distance (metres) Angle of elevation (degree) 950 7 1500 12 1800 15 3300 20

I think I don't need to use information about 900 metres because the path starts from elevation of 1000 metres. I imagine the distance will be the hypotenuse of a triangle and the height of a certain location will be the "opposite" part of a right-angled triangle (the height is the side in front of angle of elevation)

But somehow I feel that the information given by the question is not enough to determine the lower and upper limit of angle of elevation at the end of the section (I think the question asks about the angle of elevation at point on top of the mountain)

Is it correct that the information is not enough? Or am I missing something?

Thanks

Find the lower and upper limit of angle of elevation at the end of the section
Is that really what the question says, or does it ask for the upper limit of the elevation at the end of the section?

(I think the question asks about the angle of elevation at point on top of the mountain)
Well you wrote that it asks about the end of the section, not the top of the mountain. Which is it?

But somehow I feel that the information given by the question is not enough to determine the lower and upper limit of angle of elevation at the end of the section

Is it correct that the information is not enough? Or am I missing something?
Assuming that you correct the points above then there is an obvious and reasonable assumption that you need to make. What do you think that is?

songoku
Is that really what the question says, or does it ask for the upper limit of the elevation at the end of the section?
Ah my bad, I am sorry. I re-read the question and it asks the limit for the elevation, not the limit of the angle of elevation.

Well you wrote that it asks about the end of the section, not the top of the mountain. Which is it?
The question does write "at the end of the section" and I interpret it that the end of the section is at the top of the mountain. But after reading your post, it seems that I misunderstood the question.

So there is a path along the mountain that starts from a height of 1000 m above sea level. Then the question gives data about the elevation of a certain location of the path and the end of the section is at angle of elevation of 20o. The question asks to find the lower and upper limit of the elevation at this specific angle

Is my interpretation now correct?

Assuming that you correct the points above then there is an obvious and reasonable assumption that you need to make. What do you think that is?
The distance given in the table is assumed to be the "slant distance" (not vertical and not horizontal) and is a straight line, all measured from the base of the mountain.

Is this the assumption you mean?

Thanks

Is this the assumption you mean?
No. The question tells you exactly what the angle is at 1,500m and at 1,800m but in order to answer the question you have to make an assumption about the upper and lower bounds on the angle at, for instance, 1,650m. In the absence of any other information there is only one sensible assumption you can make, can you see what that is?

songoku
No. The question tells you exactly what the angle is at 1,500m and at 1,800m but in order to answer the question you have to make an assumption about the upper and lower bounds on the angle at, for instance, 1,650m. In the absence of any other information there is only one sensible assumption you can make, can you see what that is?
I am sorry, I don't think I understand your hint.

What I can think about the lower and upper bounds on the angle at 1650 m is that the lower bound would be 12o and the upper bound will be 15o (based on the data on the table)

So going along with that, the lower bound of the angle at 3300 m will be 15o and the upper bound will be 20o? With assumption that the path is something like a monotonic increasing function?

And you mean the path need not to be a straight line? What I had in my mind previously is to used trigonometry to find the elevation (to be more precise, I planned to use sin of angle of elevation = height / distance) so that's why I assumed the distance measured is straight line distance from starting point

If the path need not to be straight line, then the angle at certain location is the angle between the tangent line of that location and horizontal?

Thanks

I am sorry, I don't think I understand your hint.
I think you do , because this is it:
With assumption that the path is something like a monotonic increasing function?
Yes!

What I can think about the lower and upper bounds on the angle at 1650 m is that the lower bound would be 12o and the upper bound will be 15o (based on the data on the table)
Yes!

So going along with that, the lower bound of the angle at 3300 m will be 15o and the upper bound will be 20o?
Again yes!

And you mean the path need not to be a straight line?
We are required to find the paths with the lowest and highest possible altitudes at the point 3,300m from the start: these paths will have straight sections with 'kinks' at each measured point.

What I had in my mind previously is to used trigonometry to find the elevation (to be more precise, I planned to use sin of angle of elevation = height / distance) so that's why I assumed the distance measured is straight line distance from starting point
That makes sense, but you need to apply that piecewise to each section for which you have data.

If the path need not to be straight line, then the angle at certain location is the angle between the tangent line of that location and horizontal?
If the lines have kinks there will not be a tangent. It might help to draw it on paper: remember the angle at 1,500m must be exactly 12 degrees but at 1,500.0000001m it could have a maximum value of 15 degrees.

songoku
We are required to find the paths with the lowest and highest possible altitudes at the point 3,300m from the start: these paths will have straight sections with 'kinks' at each measured point.

Based on your hint, I image the path will be something like that, where:
AB = 950 m and θ1 = 7o
AB + BC = 1500 m and θ2 = 12o
AB + BC + CD = 1800 m and θ3 = 15o
AB + BC + CD + DE = 3300 m and θ4 = 20o

Is that correct?

If that is correct, I think I will be able to answer the question. I just need to find the elevation at each point (from B to E).

HB = 1000 + 950 sin 7o then I just repeat the process to get all the elevation and the lower bound will be the elevation at point D and upper bound will be the elevation at point E.

Is my idea correct?

Thanks

Is my idea correct?
Yes, that should give you the greatest possible elevation, what about the least?

songoku
Yes, that should give you the greatest possible elevation, what about the least?
The greatest possible elevation would be:
HE = 1000 + 950 sin 7o + 550 sin 12o + 300 sin 15o + 1500 sin 20o

and the least possible elevation would be:
hE = 1000 + 950 sin 7o + 550 sin 12o + 1800 sin 15o

Is that correct?

Thanks

The angle of elevation at 950m along the path is 7°, but where does it say it must be that at 949.999m?

songoku
The angle of elevation at 950m along the path is 7°, but where does it say it must be that at 949.999m?
Would it be like this:

The least possible elevation = 1000 + 950 sin 0o + 550 sin 7o + 300 sin 12o + 1500 sin 15o

Thanks

pbuk
Thank you very much for all the help and explanation pbuk