Finding the lower and upper limits for angle of elevation

[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

 

[pbuk]

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 20^o. 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

 

[pbuk]

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 12^o and the upper bound will be 15^o (based on the data on the table)

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

 

[pbuk]

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 12^o and the upper bound will be 15^o (based on the data on the table)

Yes!

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

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 = 7^o
AB + BC = 1500 m and θ₂ = 12^o
AB + BC + CD = 1800 m and θ₃ = 15^o
AB + BC + CD + DE = 3300 m and θ₄ = 20^o

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).

H_B = 1000 + 950 sin 7^o 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

 

[pbuk]

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:
H_E = 1000 + 950 sin 7^o + 550 sin 12^o + 300 sin 15^o + 1500 sin 20^o

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

Is that correct?

Thanks

 

[pbuk]

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 0^o + 550 sin 7^o + 300 sin 12^o + 1500 sin 15^o

Thanks

 

[songoku]

Thank you very much for all the help and explanation pbuk