Question

Water is draining at a rate of 2 cubic feet per minute from the bottom of a conically shaped storage tank of overall height 6 feet and radius 2 feet. How fast is the height of water in the tank changing when 8 cubic feet of water remain in the tank? Include appropriate units in your answer. (volume of cone: V=(1/3)(pi)r^2) your answer may be in terms of pi

Answer 1

is this related rates problem? yes it is I love these

Answer 2

Yes it is. I got an equation V=1/12(pi)h^3 but idk what to do with it

Answer 3

okay I am just going to quickly list our rates and our constants

Answer 4

v' = 2, h = 6 , r=2, v = 8, and we are looking for h' = ?

Answer 5

your volume of a cone should have h in it it should be v=(1/3)pir^2h I assume that was just a typo though. Anyways you use similar triangles already to get V=(1/12)pih^3 right?

Answer 6

If so you just need to know how to use implicit differentiation do you understand implicit differentiation?

Answer 7

yes i do

Answer 8

okay so just use implicit differentiation on the equation you got and I can tell you if it is correct sound cool?

Answer 9

what do i use it on though
like whats the equation i need to use it on

Answer 10

because the V=1/12pih^3 is only one sided

Answer 11

you use it on the one you got with the h^3

Answer 12

well iguess i could use it anyway, but i know with inplicit differentiation i put a dy/dx wherever i differientiate a y, but what would i use that for here?

Answer 13

use the prime symbol ' it will be less messy you want to use implicit differentiation because you are looking for h' and you have v' so when you do implicit differentiation you will get both these values in the equation and you will be able to solve for h' know what I mean?

Answer 14

v'=(pi/4)h^2

Answer 15

so since im using implciit i make 0=(pi/4)h^2 and solve for h?

Answer 16

you forgot about h' but you almost have it

Answer 17

you need h' so you will have an equation with v', h, and h'

Answer 18

you are very close just left out one step remember the dy/dx except we are using h'

Answer 19

oh ok
but if its 0=(pi/4)h^2 dh/dx then if i multiply or divide or whatever its always 0

Answer 20

wait you are starting with v' remember v is not one of the constants

Answer 21

or do i plug in the 2 for v'?

Answer 22

Exactly =)

Answer 23

so 2=(pi/4)h^2 dh/dx

Answer 24

right =) keep going

Answer 25

i got 8/pih^2

Answer 26

right and then you also have the h so you plug in h which is? and then you get h' at that moment.

Answer 27

yea so i got 1/4pi

Answer 28

im guessing thats the answer? plus like a per something per something because its asking how fast?

Answer 29

That should be right the only problem is I just realized the h you have is for the total height of the tank not the height at that moment which doesn't appear to be given and the radius wasn't given. You need to solve for h do you know how to do that?

Answer 30

oh so instead of doing implciit diferentiation

Answer 31

no you did everything right so far

Answer 32

i have to solve for h instead?

Answer 33

you just need to solve for h at the given volume and then plug that h into the formula that differentiated you did need to do the implicit differentiation.

Answer 34

ok im a little confused.. my final answer sofar is 1/4pi. Am i suppsoed to take a step back or like what am i supposed to do with this number.

Answer 35

right now you h' =(8/pi)h^2 and you don't have the h at that given time you need to solve for the h then plug that into the formula. You need to use V=(1/12)pih^3 where v=8 because the volume at the time we are looking for our height is 8cubic feet. Know what I mean?

Answer 36

***right now you have h' =(8/pi)h^2

Answer 37

so 8=(1/12)pih^3 and solve for h?

Answer 38

correct

Answer 39

Then when you have that h you need to plug it into h' =(8/pi)h^2

Answer 40

when solving for h you should get something like this h = cubedroot(8/((1/12)pi))