Swedish Math problem i got. Seems to be easy. but not quite ?

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Homework Help Overview

The discussion revolves around a problem involving optimal seating in a movie theater to achieve the best viewing angle of a screen. The scenario includes specific dimensions of the screen and seating arrangement, as well as the angle of incline of the seats. Participants are exploring the geometric relationships and calculations necessary to determine the ideal position along the incline.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the importance of eye level in relation to the center of the screen and question how to prove that this position maximizes the viewing angle. Various mathematical approaches are suggested, including the use of trigonometric identities and differentiation to find maximum angles.

Discussion Status

There is active engagement with multiple participants offering different perspectives and methods for approaching the problem. Some participants express confusion about the proposed methods, while others are attempting to clarify their understanding and explore new ideas. No consensus has been reached, but several productive lines of inquiry are being pursued.

Contextual Notes

Participants mention diagrams and sketches to aid in their understanding, indicating that visual representation is a key aspect of the discussion. There are references to specific angles and distances that may require further clarification or assumptions about the setup.

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Swedish Math problem i got. Seems to be easy. but not quite...?

This is a problem i chose because i thought it would be interesting as well as easy, however, i should have known better than to choose the last question in my maths book as my "essay" question. I have done it and i can't see how my method is wrong in any way. The answer i get is not realistic hence, something must be wrong. I will not show my calculations as it would take too long.

Question:
You are in a movie theatre. The screen is 8 m high and 2 meters from the ground as well as 3 meters from the first row of seats. All the seats are on an incline att an angle of 22 degrees. When you sit on a chair your eyes are 1 meter from the incline. Where on the incline are you to sit so that you will have the best viewing angle?
This is how it looks like and how i have thought a bit!
https://docs.google.com/drawings/d/1Ry5s05KkhwvfF1u8S2aAqYmMrkE92XgxNOYC-W7fS44/edit

The text translates to "what value of "h" will give the best possible viewing angle? i.e how far upp the incline must you walk?"
Answer i got was 2.56 meters. Any help would be appreciated.
 
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Would you not want your eye to be level with the centre of the screen?
 
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That seems reasonable. However i would need to prove that having the eye level at the centre gives the highest angle, something which i have no clue how to do. Should i use a^2=b^2+c^2 - 2bc(cos(A)) ?
 
Cyrus1 said:
Should i use a^2=b^2+c^2 - 2bc(cos(A)) ?
That formula could be useful in some way but first you have to find out what you want to calculate.

Did you draw a sketch?

Jilang said:
Would you not want your eye to be level with the centre of the screen?
In the real world, probably. Here, not.
 
mfb said:
That formula could be useful in some way but first you have to find out what you want to calculate.

Did you draw a sketch?

I want to calculate the highest possible value for Z, which you can see here. I differentiated a bit at the end when i got a function of h. I probably forgot to mention that it is necessary to have a function f(h)=Z
https://docs.google.com/drawings/d/1Ry5s05KkhwvfF1u8S2aAqYmMrkE92XgxNOYC-W7fS44/edit
 
mfb said:
That formula could be useful in some way but first you have to find out what you want to calculate.

Did you draw a sketch?

In the real world, probably. Here, not.
Mfb, please remind me never go and see a movie with you, lest I end up with a stiff neck, lol!
 
Cyrus1 said:
I want to calculate the highest possible value for Z, which you can see here. I differentiated a bit at the end when i got a function of h. I probably forgot to mention that it is necessary to have a function f(h)=Z
https://docs.google.com/drawings/d/1Ry5s05KkhwvfF1u8S2aAqYmMrkE92XgxNOYC-W7fS44/edit

If we are allowed stiff necks then...
Call the distance below the centre of the screen a, split z into z1 and z2 above and below the horizontal axis, derive expressions for z1 and z2. Then use...
Tan z = tan z1 + tan z2/(1-tan z1tan z2)
Then I would Differentiate to find the maximum of tan z.
Is that what you did?
 
Jilang said:
If we are allowed stiff necks then...
Call the distance below the centre of the screen a, split z into z1 and z2 above and below the horizontal axis, derive expressions for z1 and z2. Then use...
Tan z = tan z1 + tan z2/(1-tan z1tan z2)
Then I would Differentiate to find the maximum of tan z.
Is that what you did?

This is not what i did. Interesting, i will try it out and get back to you. Thanks!
 
Jilang said:
If we are allowed stiff necks then...
Call the distance below the centre of the screen a, split z into z1 and z2 above and below the horizontal axis, derive expressions for z1 and z2. Then use...
Tan z = tan z1 + tan z2/(1-tan z1tan z2)
Then I would Differentiate to find the maximum of tan z.
Is that what you did?

Ok i am lost. I have no idea what you did there.
 
  • #10
Jilang said:
Would you not want your eye to be level with the centre of the screen?

I doubt this is what is wanted. They probably want the position where the screen subtends the greatest angle at the eye.
 
  • #11
haruspex said:
I doubt this is what is wanted. They probably want the position where the screen subtends the greatest angle at the eye.


Exactly! Any ideas?
 
  • #12
Cyrus1 said:
Exactly! Any ideas?

What do you know about angles subtended by chords of a circle?
 
  • #13
I know the basics. I might post my method tomorrow as i just can't figure out what is wrong with it, but i am interested to know how you are thinking!
 
  • #14
Cyrus1 said:
I know the basics. I might post my method tomorrow as i just can't figure out what is wrong with it, but i am interested to know how you are thinking!
Draw a circle passing through the top and bottom of the screen and through a position on the seating ramp. What position maximises the angle?
 
  • #15
I drew it all very detailed and the problem is that the circle only touches a part at the bottom of the extended hypotenuse!
 
  • #16
I got that the eye level needs to be 63 cm lower than the bottom of the screen. We must be looking for a seat really near the front, less than 4m away from the screen. Stiff neck and nausea!
 
  • #18
Jilang said:
Sorry the description wasn't great. I have made a diagram.
http://tinypic.com/view.php?pic=igdb28&s=8#.UzmpDDK9KSM

Now i understand! Very nice, thanks ! however, i don't see how this works if the triangle in red is representing the angle of sight. in my head none of the equations would work out. Would appreciate an explanation! THANKS!
 
  • #19
Ah, yes that occurred to me too, but I think the tan(a+b) formula must still hold for negative angles. The blue lines are what I drew initially setting up the problem and the red line are the solution I got by maximising tan (a+b).
 

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