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        riddles >> putnam exam (pure math) >> A High School level Problem (Interesting)
        
(Message started by: daemonturk on Jul 27^th, 2009, 3:47am)

Title: A High School level Problem (Interesting)
Post by daemonturk on Jul 27^th, 2009, 3:47am

A straight track on a hillside makes an angle of 50 degrees with the line of greatest slope and an angle of 11 degree 28 minutes with the horizontal. Find the angle the hillside makes with the horizontal.

Cookies goes to the shortest and simplest solution. :D

Title: Re: A High School level Problem (Interesting)
Post by Grimbal on Jul 27^th, 2009, 4:24am

Are the 50° measured horizontally (as on a map) or in 3D?

Title: Re: A High School level Problem (Interesting)
Post by daemonturk on Jul 27^th, 2009, 8:51am

For that particular angle, 3D.

Title: Re: A High School level Problem (Interesting)
Post by Grimbal on Jul 27^th, 2009, 9:13am

[hide] 18°? [/hide]

Title: Re: A High School level Problem (Interesting)
Post by daemonturk on Jul 27^th, 2009, 9:21am

Correct. But working?

Title: Re: A High School level Problem (Interesting)
Post by Grimbal on Jul 27^th, 2009, 9:35am

[hide]Let's call alpha = 11°28, beta = 50°, and gamma the hill slope.

If we take a unit length l of track and call h is the altitude difference, then
h = l·sin(alfa)

On the other side, if d is the distance along the greatest slope that has the same difference, we have
d = l·cos(beta)
and
h = d·sin(gamma)

All this together gives
sin(gamma) = h/d = sin(alfa)/cos(beta).

It would be clearer with a picture that shows where the right angles are.
[/hide]

Title: Re: A High School level Problem (Interesting)
Post by daemonturk on Sep 1^st, 2009, 4:21am

Ya.