Question

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

Answer 1

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

Answer 2

@iop360 Do you have the Volume formula?

Answer 3

of a regular cone, yes

Answer 4

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

Answer 5

No this one, have 2 bases!

Answer 6

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

Answer 7

R is top radius?

Answer 8

b and B represent the different volumes?

Answer 9

how did you derive this formula

Answer 10

oh ok

Answer 11

ill try it. thanks!

Answer 12

hmm

Answer 13

have you tried it?

Answer 14

im getting a wrong answer

Answer 15

i got .383...m/min

Answer 16

answer is supposed to be 10.3 cm/min

Answer 17

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

Answer 18

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

Answer 19

what did you get?

Answer 20

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

Answer 21

ok yeah thats what i get now

Answer 22

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

Answer 23

dh/dt beside it of course

Answer 24

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

Answer 25

0.395...m^3/min

Answer 26

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

Answer 27

did you get 0.395...m/min

Answer 28

doesnt that still result in 39.5cm/min then?

Answer 29

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

Answer 30

it might be a solution, but i dont get it

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

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

Answer 35

!!! i think i got it

Answer 36

forgot the pi from the above forumla

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

im going to calculate it again..

Answer 41

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

Answer 42

hm ok

Answer 43

thanks

Answer 44

im going to try it with different values

Answer 45

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

Answer 46

uh oh

Answer 47

ill try it with r/R

Answer 48

annnnnnd were back to 21pi/27 h^3

Answer 49

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

Answer 50

and yet i got close to a right answer

Answer 51

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

Answer 52

Yes, we calculate the instant rate when h = 1

Answer 53

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

Answer 54

*learning* ._.!

Answer 55

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

Answer 56

ok i will

Answer 57

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?

Answer 58

answer: 1.4cm/min

Answer 59

alright

Answer 60

i get 1.9 ..hmmm

Answer 61

it could be a glitch with the thing, maybe.

Answer 62

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

Answer 63

thanks for the help

Answer 64

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

Answer 65

thanks!

Answer 66

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

Answer 67

thanks!

Answer 68

@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?

Answer 69

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