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Math Problem - Help?

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RJD2
Posts: 3532 - Karma: 921
I am really struggling with this math problem for a project I'm working on at work. Anyone???



Dec 22 2011 11:26AM
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ride-rider
Posts: 1761 - Karma: 20
Uhhh, ummmmm, uhhhhhhhhhhhhhhhhh, uh.  Oh wait!  Uhhhhhhhhh.  Ummmmmm.  I am not sure?
Dec 22 2011 11:28AM
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feihlination
Posts: 8356 - Karma: 1,084
in the honor of a recent thread, write that shit in metric system.

and i would just try to formulate a function and solve it by analysing it
Dec 22 2011 11:34AM
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imajew
Posts: 324 - Karma: 22
find the volume of cone 3.75 base by 6 height.
take away the volume of cone 2.5 base by 6 height.

now you have volume of the cup.

volume of a cone is 1/3 base(area) height

so half the volume of the cup, then do (half volume of cup) = 1/3 base(area) height




Gimme a few more minutes to think about this. Theres a lot of shit going on here.


Dec 22 2011 11:36AM
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RJD2
Posts: 3532 - Karma: 921
all i've got so far is

πR(^2)h = volume
(π*3.75)(π*2.5)*6=volume

Now i'm lost
Dec 22 2011 11:38AM
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imajew
Posts: 324 - Karma: 22
okay, my fucking head hurts, i give up.
Dec 22 2011 11:38AM
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RJD2
Posts: 3532 - Karma: 921
and i actually did the (π * circumference) there.. that's not even right, so..... damnit
Dec 22 2011 11:42AM
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Dyar
Posts: 622 - Karma: 35
I'll help you give me a few minutes!
Dec 22 2011 11:42AM
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ride-rider
Posts: 1761 - Karma: 20
A for effort
Dec 22 2011 11:42AM
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feihlination
Posts: 8356 - Karma: 1,084
i show you in a minute
Dec 22 2011 11:43AM
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Dyar
Posts: 622 - Karma: 35
Is 6 inches the height or the actual length of the side?
Dec 22 2011 11:45AM
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Dyar
Posts: 622 - Karma: 35
Nvm I'm retarded^

Dec 22 2011 11:49AM
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joris.blanc
Posts: 914 - Karma: 6,584
Use this formula to find the total volume V= pi H/3 (R²+r²+Rr) H is the 6in, R is (3.75/2)in and r is (2.5/2)in.

Than take the total volume divide it by 2

Now with the use the same formula as before and make it equal to the half volume, now you have a formula with 2 unknown variables (H' and R') keep the r as it is (2.5/2)in.

With the triangles you can link the H and R together
you will find something like R' = 1/6 ((R-r) h)-r

Than just replace the result in the formula and you can find the H'

Dec 22 2011 11:59AM
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DayMan
Posts: 4674 - Karma: 8,285
Holy tits
Dec 22 2011 12:02PM
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Dyar
Posts: 622 - Karma: 35
H=2.423 approx
Dec 22 2011 12:24PM
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joris.blanc
Posts: 914 - Karma: 6,584
haha if anyone asks if 2.423 is from the top or the bottom of the cup, I will start laughing !!!!
Dec 22 2011 12:27PM
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imajew
Posts: 324 - Karma: 22
...it can't be, it has to be greater than 3 at least because the cup's wider at the top than the bottom. its going to be closer to 4 than 2.
Dec 22 2011 12:27PM
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BootySpunk.
Posts: 8148 - Karma: 83
I would do this by integration. But I don't think it is supposed to be done that way.

Dec 22 2011 12:34PM
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Dyar
Posts: 622 - Karma: 35
V= pi H/3 (R²+r²+Rr)

V=pi*6/3((3.75/2)^2+1.25^2+1.25*3.75/2)=46.633 approx

(1/2)*46.633=pi(6-h)/3(1.25^2+1.25r+r^2)

Solve for h

H=something really ugly

Plug the h into:

23.317=pi*h/6(

Son of a bitch I fucked it up
Dec 22 2011 12:36PM
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Errl
Posts: 2082 - Karma: 451
not only does he take bitchin photos....
Dec 22 2011 12:40PM
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Dyar
Posts: 622 - Karma: 35
Actual answer
H=3.811 aprox
Dec 22 2011 12:40PM
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Dyar
Posts: 622 - Karma: 35
But yeah do what jorisblanc said except I solved for r in terms of h and then plugged that into the rs and solved for h.
Dec 22 2011 12:45PM
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RJD2
Posts: 3532 - Karma: 921
Thanks for all your help guys. We are still working on the paper formula here, but we actually own several of these cups in the office...After several hours of failed math attempts and trying these formulas, we decided to take one cup and fill it to the brim, then empty it into the other cup until they were equal. Our approx. measurement is 3.7inches from the bottom
Dec 22 2011 12:46PM
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joris.blanc
Posts: 914 - Karma: 6,584
You probably could do it by integrating, but I'm pretty sure that the OP has already enough trouble with this problem as it is ;-)
Dec 22 2011 12:51PM
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joris.blanc
Posts: 914 - Karma: 6,584
Just wondering, why do you want to find the center volume of a cup ???
Dec 22 2011 12:53PM
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RJD2
Posts: 3532 - Karma: 921
Good question - I had to create a design/theme for some plastic cups we are giving away internally to several departments in the company. We have our company logo on one side, and then I came up with these theme for the reverse :)

Dec 22 2011 1:00PM
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joris.blanc
Posts: 914 - Karma: 6,584
Okay cool !!
Dec 22 2011 1:02PM
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RJD2
Posts: 3532 - Karma: 921
Thanks for the help everyone. + to allllll
Dec 22 2011 1:06PM
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mizzike
Posts: 1386 - Karma: 251
This is a lot more complicated than it first seems..

I found this nifty website that calculates the volume of a truncated cone..

http://keisan.casio.com/has10/SpecExec.cgi

The site also gives the formulas used in their algorithm. I'm pretty confident that with this information we can find the "midpoint".

In your drawing you say that you have the circumferences of the the two circles that make up the top and bottom of the cup, but i'm assuming that you meant to say diameter because of the way you've drawn your dimensions. If you do actually mean circumference you can easily solve for the radiuses using the formulas:

Circumference = π*d and Radius = d/2

Anyway, back to the actual problem..

When you punch in the dimensions of your cup it stays that the volume is 46.63
Now divide that number by two and and set it equal to the volume formula they provide and sub in all of the information you know..

This gives us an equation that looks like this:

23.31 = (1/3)*π*(R1^2+ R1(1.25) + (1.25)^2)*H

Where:
23.31 is half of the total volume
R1 is the unknown radius of the circle that is the top of the new "half cup"
R2 is the radius of the bottom of the cup
and H is the unknown height.

Now it appears that we cannot solve this for H because we also have R1 as an additional unknown, making 2 unknowns and only one equation. However, because of the cups conical shape, we can relate R1 to H using the principle of similar triangles.

If you look back at the original drawing, we see that the slope of the side of the cup can be described by the difference in the size of the radiuses on it's ends and the total distance over which they change.

Slope = (1.87-1.25)/6
Slope = 0.1 (approximately)

So now that we know the slope we can say that the radius at the middle is just the height at the middle (H) times the slope (.1) plus the radius at the bottom

or written mathematically as..

R1 = .1*(H) + 1.25

Now we can actually solve for H!
Here's the equation I came up with the 2 unknowns..

23.31 = (1/3)*π*(R1^2+ R1(1.25) + (1.25)^2)*H

Now sub (.1*H+1.25) in anywhere R1 appears and we get..


23.31 = (1/3)*π*[(.1*H+1.25)^2 + (.1*H+1.25)*(1.25)+(1.25)^2]*H

^ This is literally a nightmare to try and solve so i'm just gonna use wolframalpha.com

http://www.wolframalpha.com/input/?i=solve+23.31+%3D+%281%2F3%29*%28pi%29*%5B%28.1*H%2B1.25%29%5E2+%2B+%28.1*H%2B1.25%29*%281.25%29%2B%281.25%29%5E2%5D*H+for+H

which says that H = 3.607



Dec 22 2011 1:26PM
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Anathema
Posts: 21048 - Karma: 5,059
its actually really easy to get the height of the cone if it were to be continued.

3.75 to 2.5 is a difference of 1.25. so basically, every 1.25 inches the diameter shrinks, it gains 6 inches in length.

at the 3.75 mark it has 0 height
at the 2.5 mark it has 6 height
at the 1.25 mark it has 12 height
at the 0 mark (tip of the cone) it has 18 height

just using a little intuition, you've got a large chunk of the problem down.

now, knowing the total volume of the cylinder (since you have r = 3.75 and h = 18) and the volume of the tip of the cylinder (r = 2.5 and h = 12), you can calculate the volume of the picture shown (which is about 46.64). since the midpoint represents the height at which both volumes are equal, you know that they both have to be at a volume of 23.32.


we know this much already, having barely done any complicated math or setting up equations with variables that will help us determine the height at which the volume of either the top half or the bottom half will be 23.32.

now, it's just a question of doing what ^^ that guy said and solving for the midpoint.
Dec 22 2011 2:12PM
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reeferbrigade
Posts: 526 - Karma: 37
take the water in the cup and pour it into a box.
Dec 22 2011 3:25PM
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