Re: gas tank problem

[Walt Mankowski <waltman@pobox.com>]

It's past my bedtime, too, but one thing I saw right away is that
you're solving for the wrong value.  You want to find h when Agas is
1/4 the area of the entire circle.

Walt


On Tue, Dec 10, 2002 at 01:21:49AM -0500, Jeff Weisberg wrote:
> 
> 
> I knew once I sat down with a pencil and paper
> the answer would become "obvious".
> 
> it can be solved without calculus, using just basic
> trigonometry.
> 
> the hard part is finding the area bounded by
> the circle and the surface of the gas. multiplying
> by the length of the cylinder yields volume.
> 
> 1:	draw a circle.
> 
> 2:	draw the surface of gas. a chord from one
> 	point on the edge of the circle to another.
> 	call the length of this chord "2 * Y"
> 	this length is currently unknown.
> 
> 3:	draw lines from these 2 points to the center
> 	of the circle. they go from the edge of the circle
> 	to the center--radii, this length is known, call it "R"
> 
> 4:	call the angle between these 2 radii, "theta"
> 	this is unknown
> 
> 5:	we know the height of gas from our dipstick.
> 	we call this height "h".
> 
> 6:	the distance of the gas surface to the center
> 	of the circle is given by:
> 		X = R - h
>   	
> 7:	make a poor attempt at an ascii drawing:
>                  .---.
>               .-'     '-.                    
>             .'           '.               . theta
>            /               \             /|\
>           |                 |       R  /  |  \ R
>           |        o        |        /    |X   \
>           \       /|\       /      /      |      \
>            '   R /x| \    .      /________|________\
>             '-. /--|-y\ .-'        y      |h     y    
>                '- h|_._'                __|__
> 
> 
> 8:	notice that area we are trying to find "Agas"
> 	is the area of a pie shaped wedge minus the
> 	area of a triangle:
> 		Agas = Apie - Atri
> 
> 
> 9:	simple trig:
> 		Apie = 1/2 * R^2 * theta
> 		Atri = X * R sin theta/2
> 
> 10:
> 	more trig:
> 		theta = 2 * cos-1( X/R )
> 
> 11:
> 	we know X, we know R. we now know theta.
> 
> 12:	we now can calculate Apie, Atri, and Agas.
> 
> 13:	simplify the equations in terms of our known values:
> 
> 		Agas = R^2 * cos-1( R-h / h ) - (R - h) * sqrt( 2 R h - h^2 )
> 
> 
> 14:	I had a beer at dinner, and it is after my bedtime.
> 	check my math before deciding to head into the desert
> 	without filling up.
> 
> 
> 	
> 	--jeff
> 
> 
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