Fuel Gage Problem

[Mano]

The problem with that would be, either he would have to run the tank completely dry (which would be bad, and difficult to time), or there would be some undetermined amount of gas left in the tank.

[JTG51]

That is a good point. But, since we only seem to be interested in measuring quarters of a tank, it doesn't need to be extremely accurate.

Of course, I don't think any of this matters. I don't think Jim is looking for a common sense solution, he's looking for the math. At least I think so.

I still like our ideas better. [img]/forums/images/icons/smile.gif[/img]

[irchans]

After reading JTG51's post, I wondered if there were solutions which did not require trig or calc. I thought of two. Maybe others can think of more.


Solution 1: Monte Carlo

a) Genereate 1 million points in a 20 by 20 area
b) Remove all the points that are more than 10 from the center to get a disk of points
c) Sort the points by height
d) Choose the point that is 1/4 of the way through the list

I did this with a simple program and got 5.95613 on the first run and 5.96997 on the second run.


Solution 2 (Integration with out calculus):

a) Split the 20 inch circle into horizontal strips of height 0.01 inches.
b) Approximate the area of each strip.
c) Add up areas starting from the bottom until you reach 78.54 square inches ( 1/4 of the area ).

Using excel, this method gave 5.97 inches.

This example shows that computer skills can sometimes serve as a subsitute for math skills.

[Homer]

Very nice irchans...

[JTG51]

Wow, very creative irchans. I like both ideas.

Although, I don't think anyone without strong math skills could have come up with either method, or solved them.

This is a great example of the fact that there's always multiple ways to solve any problem. A little thought beforehand can often times save you a lot of thought later.

[Mike Haven]

draw a circle of radius 10cm to represent the cross-section

draw a chord across below the centre to represent the liquid level

join the ends of the chord to the centre to form a triangle with angle A

the area of the liquid equals the area of the sector minus the area of the triangle

using radians as angle measure this is calculated as (A x radius squared / 2) minus (1/2 x radius squared x sinA)

this comes to 50A - 50sinA and is known to be a quarter of the tank cross-section or pi x radius squared / 4 or 25 x pi

this equation, because of the mixture of trig and non-trig can only be solved by calculator iteration (don't forget we are working in radians) and the answer is 2.31 radians

half this to get the half angle of the triangle and use 10 x cosine1.155 (calculator on radians) to find the vertical distance of the liquid level below the middle

subtracting this answer from 10 gives you the dipping measurement of 5.96

[Munga30]

You don't have to run it dry.

Fill the tank to 10 inches, which you know is half a tank.

THEN add the 1/4 tank, measured at the pump in gallons.

THEN measure with the stick and subtract 10 inches.

[Phat Mack]

</font><blockquote><font class="small">In reply to:</font><hr />
Solution 2 (Integration with out calculus):

a) Split the 20 inch circle into horizontal strips of height 0.01 inches.
b) Approximate the area of each strip.
c) Add up areas starting from the bottom until you reach 78.54 square inches ( 1/4 of the area ).


[/ QUOTE ]

How did you approximate the area of each strip?