A large container has the shape of a frustum of a cone with top radius 8m, bottom radius 2m, and height 6m. The container is being filled with water at the constant rate of 2.9m^3/min. At what rate is the level of water rising at the instant the water is 1m deep?
Answer: 10.3cm/min

- anonymous

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- anonymous

\[\frac{ dV }{ dt } = +2.9, \frac{ dH }{ dt }=?\]

- anonymous

@iop360 Do you have the Volume formula?

- anonymous

of a regular cone, yes

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## More answers

- anonymous

\[V = \frac{ 1 }{ 3 }(\pi)r^2h\]

- anonymous

No this one, have 2 bases!

- anonymous

i think what we are supposed to do though is make it into two cones though

- anonymous

|dw:1353202550485:dw|

- anonymous

R is top radius?

- anonymous

b and B represent the different volumes?

- anonymous

how did you derive this formula

- anonymous

oh ok

- anonymous

ill try it. thanks!

- anonymous

hmm

- anonymous

have you tried it?

- anonymous

im getting a wrong answer

- anonymous

i got .383...m/min

- anonymous

answer is supposed to be 10.3 cm/min

- anonymous

oh wait i think i made an error.. let me recalculate

- anonymous

my expression for V is:
\[V = [\frac{ 16 }{ 27 }(\pi)h^3 + \frac{ 4}{ 27 }(\pi)h^3 + \frac{ 1 }{ 9 }(\pi)h^3]\]

- anonymous

what did you get?

- anonymous

oh wait.. 1/27, not 1/9

- anonymous

ok yeah thats what i get now

- anonymous

taking its derivative, you get
\[\frac{ 7 }{ 3 }(\pi)h^2\]

- anonymous

dh/dt beside it of course

- anonymous

(2.9)/(7pi/3) = dh/dt after you plug h =1, which does nothing

- anonymous

0.395...m^3/min

- anonymous

i think we did it right, not sure why its resulting in the wrong answer

- anonymous

did you get 0.395...m/min

- anonymous

doesnt that still result in 39.5cm/min then?

- anonymous

try reading this http://www.askmehelpdesk.com/mathematics/related-rates-frustum-cone-352086.html

- anonymous

it might be a solution, but i dont get it

- anonymous

hmm is there anything extra in the formula you may have forgotten

- anonymous

http://www.youtube.com/watch?v=1v1Pp-lJSKY this video ?

- anonymous

http://jwilson.coe.uga.edu/emt725/Frustum/Frustum.cone.html
this link doesnt have a square root in the formula

- anonymous

bleh, the formula without the sq.root doesnt work either...

- anonymous

!!! i think i got it

- anonymous

forgot the pi from the above forumla

- anonymous

http://jwilson.coe.uga.edu/emt725/Frustum/Frustum.cone.html

- anonymous

i got 10.1 though...calculator didnt take exact values

- anonymous

\[V = \frac{ (\pi)h }{ 3 }[R^2 + Rr + r^2]\]

- anonymous

im going to calculate it again..

- anonymous

im getting 10.05... cm/min
that is slightly off

- anonymous

hm ok

- anonymous

thanks

- anonymous

im going to try it with different values

- anonymous

oh wait...i used base for r in that last formula

- anonymous

uh oh

- anonymous

ill try it with r/R

- anonymous

annnnnnd were back to 21pi/27 h^3

- anonymous

yeah, i accidently used the B/b for that forumla instead of R/r

- anonymous

and yet i got close to a right answer

- anonymous

do we sub in h = 1 or h =6?
h = 1 is right isnt it?

- anonymous

Yes, we calculate the instant rate when h = 1

- anonymous

the other guy reading this question: are you doing this question?

- Callisto

*learning* ._.!

- anonymous

i could switch the values for the question and try it again if you want, haha

- anonymous

ok i will

- anonymous

a large container has the shape of a frustum of a cone with top radius 10m, bottom radius 6m, and height 4m. The container is being filled with water at the constant rate of 2.9m^3/min. At what rate is the level of water rising at the instant rate the water is 2m deep?

- anonymous

answer: 1.4cm/min

- anonymous

alright

- anonymous

i get 1.9 ..hmmm

- anonymous

it could be a glitch with the thing, maybe.

- anonymous

ok, im pretty sure were doing it right. imma close this down

- anonymous

thanks for the help

- anonymous

Check with your teacher about the formula!
I'll sure will message you if I find out something interesting :)

- anonymous

thanks!

- anonymous

I'm so sorry that I'm underestimate this Related Rate frustum of Cone!
Here's the correct logic:
V = π/ 3 ( R² + Rr + r² ) H
With r = 2 and R is Radius variable and H is the Height variable
-> V = π/ 3 ( R² + 2R+ 4 ) H ( 1 )
From the ratio of right triangle:
( R - 2 ) / ( 8 - 2 ) = H/ 6
--> R = H + 2 (2)
Plug (2) into (1):
=> V = π/ 3 [ ( H +2) + 2( H +2) + 4 ] H
= π/ 3 [ H³ + 6H² + 12H ]
so V' = π/ 3 [ 3H² + 12H + 12 ] H'
At H = 1:
V' = 27 π/ 3 * H'
2.9 = 9 π * H'
Thus H' = 2.9 / 9π = .1025 m/ min = 10.3 cm/ min

- anonymous

thanks!

- anonymous

@Chlorophyll one question though..
what do you do exactly during the ratios of the triangle part?
eg. say my top radius was 9, bottom was 8, and height was 5
how do you set it up exactly?

- anonymous

Ratio R and H:
( R - 8 )/ ( 9-8) = H/5
-> R = ( H/5 ) + 8

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