Question

Help with this Related Rates Problem.
Water is flowing at the rate of 5 cubic meters/min. into a tank in the form of a cone of
height 20 meters and base radius 10 meters and with the vertex in the downward
direction. How fast is the water level rising when the water is 8 meters deep?

Answer 1

v'=5, v = (1/3)pir^2h, h' = ?, Draw a diagram or just imagine a diagram and you will notice similar triangles you can use so you know that h/r = 20/10 h=2r, you see where I am going with this?

Answer 2

I see where your going, and I was just getting to that point. I did r = 1/2h though is there a reason why I should do it the other way ?

Answer 3

these problems always confuse me :(

Answer 4

no actually I just realized I should have done that haha

Answer 5

so you put 1/2h in the formula for volume and what do you get?

Answer 6

Don't feel bad related rates problems are one of the things most people taking calculus have difficulty with.

Answer 7

i did v' = 2pi/3(1/2h)(r')*h
5 = (2pi/6)(4)(r')(8)
sounds right so far ?

Answer 8

you skipped a step and the r should be gone we are trying to get rid of the r you want to sub 1/2h into the formula for volume so what (1/2h)^2? See what I mean?

Answer 9

what's (1/2h)^2 ***

Answer 10

h^2/4

Answer 11

Excellent and then you can break that into h^2*(1/4) and ithen simplify the volume formula you want to try?

Answer 12

okay wait, got a little confused. sorry but let me just retrace my steps.
I know V' = 5
next I know the formula for V = 1/3*pi*r^2*h
I take the derivative of V or do I just use V as it is in plug in h^2/4 ?

Answer 13

You wanted to eliminate r so you replace r in v=(1/3)pir^2h with (1/2h) to get v=(1/3)pi(1/2h)^2h then you distributed the exponent and got v=(1/3)pi(h^2/4)h then in order to simplify further before you do implicit differentiation you should break (h^2/4) into h^2*(1/4) to get v=(1/4)(1/3)pih^2h and no you can make it a little bit easier before you differentiate do you see what you need to do next?

Answer 14

and now**

Answer 15

I simplify further and get (pi/12)*h^3 and theeen take the derivative ?

Answer 16

yes but you need to do implicit differentiation because you are looking for h with respect to t (time) do you know how to do this?

Answer 17

I believe so. It's just the wording on the related rates and the setup that gets me.
So I Get:
5 = (3pi/12)(h^2)(h')
I'm Looking for h' and I know h = 8 from the problem.
5 = 16*pi *h'
h' = 5/16pi ?

Answer 18

omg pls be rt I almost feel kinda smart lol

Answer 19

nope, I'm wrong I think lol.

Answer 20

Yes awesome no I am pretty sure you got that right

Answer 21

OMG dude Thanks! I just evaluated the answer and got .09947... and thought that maybe that was too small.

Answer 22

No problem you solved it. Have anymore stuff you need help with?

Answer 23

Well actually I wanted to know, if maybe you can help me on the idea of setting problems up like this. Like I get this one and If I get another like this I can do, but If I had ...which is another problem I'm trying to understand:

A boat is pulled into a dock by a rope that is attached to the bow of the boat and passing
through a pulley on the dock that is 5 feet higher than the bow of the boat. If the rope is
pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when
it is 12 feet from the dock?

Answer 24

Okay the first thing I always do is look for for the rates and constants and write them down so I know what I am working with so what are the rates and constants you see here?

Answer 25

for the rates I spot by looking for something like "per second" so I got a rate of 2ft/sec constants are 5 and 12

Answer 26

I believe this is setting up a triangle and I can use Pythagorean Theorem to figure out that third meas of 13.

now this is my hard part. I don't know what variable to assign to my constants and rate of change. what part of the triangle does the rate go on ?

Answer 27

Right you would be working with a triangle. The best way to figure this out is to draw a diagram of the problem a really simple one.

Answer 28

okay so which letter do you think represents the boat? The height? The distance from the shore?

Answer 29

x would be the distance y would be the height...process of elimination I would say z is the boat, but I would not even think the boat should be represented except for maybe a point on the triangle.

Answer 30

wait back up they are all distances in the diagram sorry my wording was poor I should have said represents the boat I should have said represents the distance from the bow to wall straight ahead like not the distance from the bow to the top of the wall.

Answer 31

shouldn't have said represents the boat***

Answer 32

ahh okay. well then I think I'm learning haha. So I diagram it out and put constants next to whatever variable I want to call them ? as long as I understand what they mean so like for x I could have used a as long as i knew that ment the distance from dock to boat?

Answer 33

Correct

Answer 34

=)

Answer 35

Now, how do we figure out where to put that rate at. I know the rate is just the derivative of something, but which one ?

Answer 36

Okay the one rate you know is what?

Answer 37

2 ft/sec

Answer 38

Right and 2ft/sec represents what?

Answer 39

the boat moving towards the dock ? so its the derivative of x in your diagram ?

Answer 40

Exactly =)

Answer 41

so here I would write down x' = 2

Answer 42

okay, I see making progress. so now I know I need an equation that = x' x'(t) = 2 .....head hurting thinking ... where do I get equation from lol ?

Answer 43

No no you are thinking to hard I used to make that mistake two. You just want to set your variables and your rates then you want to find an equation that when you do implicit differentiation will give you the variables that you have the values too. Or is that what you meant? Sorry I may I have misunderstood what you said.

Answer 44

You should write out all the things you know right now with their corresponding variables.

Answer 45

to**

Answer 46

x = 12
y = 5
z = 13 from PT
x' = 2 ft/sec

Answer 47

Yes =) now what variable are you look for?

Answer 48

see hmm... It says " speed of boat when 12 ft away".... quite simply I would think the rate hasn't changed and its still 2 ft/sec...

Answer 49

Remember that is the rate of change towards in a straight line but because the distance from the shore to the bow is at an angle the rate of change is not constant. You are looking for the rate of change of one of the other varible it is either y or z which one do you think makes more sense?

Answer 50

y isn't moving. but z is pulling a boat... So i'm looking for z specifically z'?

Answer 51

Correct =) so what formula would you use? I believe you stated it earlier

Answer 52

way back from with the cone ? or am I looking for area of triangle ?

Answer 53

not from the cone it has to do with triangles think wizard of oz haha

Answer 54

sorry that reference probably confused you but you are looking for z' so you need the derivative of the formula for z look at the diagram and you should see what formula you need

Answer 55

has to be x^2+y^2 = z^2
when I did it with the cone though I plugged in something to get rid of one of the variable though.

Answer 56

wait nvm. I know what x and y are, don't need to do that.

Answer 57

I'm on to something. Take derivative of both sides,
I don't know what y' is but I know it doesnt move so it must be 0

2x(2) + 2y(0) = 2z *z'

Answer 58

solve for z' ? right, c'mon right lol ?

Answer 59

Well you don't want to set x before deriving and don't know what y' is so there is still a problem but you know y is a constant

Answer 60

you should plug your constant in and then do implicit differentiation which will help you find z'

Answer 61

do you want to try writing that out?

Answer 62

like the step before you differentiate

Answer 63

okay give me a sec

Answer 64

12^2 + 5^2 = 13^2 that step ?

Answer 65

wheres the 13 from?

Answer 66

from what we solved for z

Answer 67

The 13 should be a z but I would save that step until after differentiation you'll see why.

Answer 68

oh wait a sec I see what you did sorry my bad I would have plugged in y before differentiating and made that a zero but yeah that makes sense but we didn't solve for z yet you should have 48/2z = z' know what I mean? Now you need to find z

Answer 69

hmm can you show me the math how you got to there

Answer 70

wait nvm I get it, you didn't solve for z yet but I did and knew it was 13.

Answer 71

Okay I'm on track. I get 48/26 = 1.8462 ft/sec

Answer 72

oh my bad when you wrote 13^2 I just got confused about what you were doing yes that's correct as I'm sure you know haha

Answer 73

awesome job dood

Answer 74

Yay! but no dood lol you've been so helpful. Haven't had anyone to like talk/chat me through the problem. That helps me SO MUCH on understanding instead of just doing it / showing next step.

Answer 75

but now I have 1 more trick question. Is the answer + or - 1.8 ft/sec ? I'm just thinking about how it is "pulled in at a rate of 2ft/sec" so does that really mean -2ft/sec ?

Answer 76

well the distance is becoming less and less as the boat approaches the shore I think making it negative would be right. No problem though anytime I am on I really like math and it is good practice for me. ttyl tho I gotta go great work man =)

Answer 77

Alright, Thanks again! I need to get going myself.