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Simple dipstick problem

by pat666
Tags: dipstick, simple
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Mentallic
#37
Feb12-11, 11:29 PM
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Quote Quote by pat666 View Post
I think it is correct but given the amount of trouble I've had getting the solution you can see why I am unsure.
Yes it's correct. I was just showing you that you can tidy up
Quote Quote by pat666 View Post
[tex] V_(fluid)=(\pi((h/H)*R)^2*h)/3 [/tex] probably simplify down further.
into
Quote Quote by Mentallic View Post
Sure, [tex]V=\frac{\pi h^3R}{3H^2}[/tex]

Quote Quote by pat666 View Post
just the way the picture was drawn made me think r=h,
Oh, yeah that would be my fault, sorry

Quote Quote by pat666 View Post
which would be true if the water was at the centre.
The blue line was meant to be where the water level was at.

Quote Quote by pat666 View Post
I get that [tex] \theta=2cos^-^1(r-h)/r [/tex]
Yes that's right.

Quote Quote by pat666 View Post
not sure if thats correct because you gave me some trig info that was a bit more complex.
My trig info was wrong, ignore it. I forgot about the 2 that was going to be in front of it. What I was meant to give you was
[tex]\sin\left(2\cos^{-1}\left(x\right)\right)=2x\sqrt{1-x^2}[/tex]
It may look complex, but its purpose is simple. When you plug [itex]\theta[/itex] into the area equation [tex]A=\frac{r^2}{2}\left(\theta-\sin\theta\right)[/tex] you're going to be left with
edit: [tex]\sin\left(2\cos^{-1}\left(\frac{r-h}{r}\right)\right)[/tex] which is where you can simplify this with the equality I gave above.
You already have the answer, but it's just if you wanted to simplify things a bit more.

Markers wouldn't give full marks if you left an answer as [tex]\sin\left(\sin^{-1}\left(x\right)\right)[/tex] so I doubt they would give full marks if you left it as [tex]\sin\left(\cos^{-1}\left(x\right)\right)[/tex] either.

Quote Quote by pat666 View Post
also this will only work to the halfway point but I was thinking I would just do the reflection of the earlier dipstick points for points after the mid line.
thanks
I believe the same formula will work for the water level anywhere from 0 to 2r (the diameter of the circle) but I'll check to see.
pat666
#38
Feb12-11, 11:38 PM
P: 709
Ok thanks alot for all your help
OmCheeto
#39
Feb12-11, 11:39 PM
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The cylinder on it's side problem reduces to y=x - sin(x) in it's simplest form.
If you can solve for x, then you can solve the problem.
There is a reason they put this problem in computer science text books and never in mathematics text books.
Mentallic
#40
Feb12-11, 11:45 PM
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Quote Quote by pat666 View Post
Ok thanks alot for all your help
No worries

Quote Quote by OmCheeto View Post
The cylinder on it's side problem reduces to y=x - sin(x) in it's simplest form.
If you can solve for x, then you can solve the problem.
Can you please elaborate?

Quote Quote by OmCheeto View Post
There is a reason they put this problem in computer science text books and never in mathematics text books.
I've seen questions similar to this in maths books.
pat666
#41
Feb12-11, 11:52 PM
P: 709
This question is for yr 11 math. My solution is I believe [tex] V=L*\sin\left(\cos^{-1}\left(2\cdot\frac{r-h}{r}\right)\right) [/tex].
Mentallic
#42
Feb12-11, 11:58 PM
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Quote Quote by pat666 View Post
This question is for yr 11 math. My solution is I believe [tex] V=L*\sin\left(\cos^{-1}\left(2\cdot\frac{r-h}{r}\right)\right) [/tex].
That's not right. The formula is [tex]A=\frac{r^2}{2}\left(\theta-\sin\theta\right)[/tex] where [tex]\theta=2\cos^{-1}\left(\frac{r-h}{r}\right)[/tex]
pat666
#43
Feb13-11, 12:14 AM
P: 709
whoops I forgot the first theta, so V=L*(r^2/2(2arccos(r-h/r)-sin(2arccos(r-h/r))) pretty messy.
OmCheeto
#44
Feb13-11, 12:43 AM
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Quote Quote by Mentallic View Post
Can you please elaborate?
Perhaps the question in the book was worded differently.
All I know is that if you have a 10 gallon tank, it is not possible to place integer gallon marks on the dipstick. Unless of course you are clever enough to solve for x.


And perhaps I should give some background regarding this particular problem before someone yells at me for playing mind games.


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