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

[Cyrus1]

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.

 

[Jilang]

Would you not want your eye to be level with the centre of the screen?

 

[Cyrus1]

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)) ?

 

[mfb]

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?

Would you not want your eye to be level with the centre of the screen?

In the real world, probably. Here, not.

 

[Cyrus1]

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

 

[Jilang]

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!

 

[Jilang]

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?

 

[Cyrus1]

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!

 

[Cyrus1]

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.

 

[haruspex]

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.

 

[Cyrus1]

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?

 

[haruspex]

Exactly! Any ideas?

What do you know about angles subtended by chords of a circle?

 

[Cyrus1]

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!

 

[haruspex]

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?

 

[Cyrus1]

I drew it all very detailed and the problem is that the circle only touches a part at the bottom of the extended hypotenuse!

 

[Jilang]

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!

 

[Jilang]

Ok i am lost. I have no idea what you did there.

Sorry the description wasn't great. I have made a diagram.
http://tinypic.com/view.php?pic=igdb28&s=8#.UzmpDDK9KSM

 

[Cyrus1]

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!

 

[Jilang]

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).