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Rate of change: conical tank [Solved!]

My question

Please help me solve a rate of change problem about a conical tank wit vertex down. i dont know the equation i have to use

Relevant page

4. Related Rates

What I've done so far

I read the examples on the page, but none of them were like my one.

X

Please help me solve a rate of change problem about a conical tank wit vertex down. i dont know the equation i have to use
Relevant page

<a href="/applications-differentiation/4-related-rates.php">4. Related Rates</a>

What I've done so far

I read the examples on the page, but none of them were like my one.

Re: Rate of change problem: conical tank

Hello Ana

You will need to provide more details before I can help you.

You may get some inspiration from this page, which is also talking about a conical tank:

4. The Graph of a Function
(go to the bottom of the page.)

Regards

X

Hello Ana

You will need to provide more details before I can help you.

You may get some inspiration from this page, which is also talking about a conical tank:

<a href="/functions-and-graphs/4-graph-of-function.php">4. The Graph of a Function</a> 
(go to the bottom of the page.)

Regards

Re: Rate of change problem: conical tank

The question says the tank has diameter 3 m at the top, and is 5 m high. What is the rate of change of the height of the water at time t?

X

The question says the tank has diameter 3 m at the top, and is 5 m high. What is the rate of change of the height of the water at time t?

Re: Rate of change problem: conical tank

You need to show us your attempts at the problem.

Did you look at that page I suggested?

Hint 1: What is the formula for the volume of a cone?

Hint 2: What is the rate of flow of the water?

X

You need to show us your attempts at the problem.

Did you look at that page I suggested?

<b>Hint 1: </b> What is the formula for the volume of a cone?

<b>Hint 2: </b> What is the rate of flow of the water?

Re: Rate of change problem: conical tank

I did look at that example, but it says r = h and my question does not have that. So I got stuck.

Anyway, volume of a cone is pir2h/3

The question doesn't say what the water flow rate is.

X

I did look at that example, but it says r = h and my question does not have that. So I got stuck.

Anyway, volume of a cone is pir2h/3

The question doesn't say what the water flow rate is.

Re: Rate of change problem: conical tank

I encourage you to use the math input system, so we can read your answer more easily.

You just need to put the following in between back ticks,

V = (pi r^2 h)/2

So it looks like this:

`V=(pi r^2 h)/3`

You're right - the question doesn't give us a number for the flow rate, but we can just give it a letter, say `f` and assume it is constant at liters per minute, say.

Hint 3: How long will it take the water to flow out?

Hint 4: How do you find the rate of change of the volume?

X

I encourage you to use the math input system, so we can read your answer more easily.

You just need to put the following in between back ticks, 

<code>V = (pi r^2 h)/2</code>

So it looks like this:

`V=(pi r^2 h)/3`

You're right - the question doesn't give us a number for the flow rate, but we can just give it a letter, say `f` and assume it is constant at liters per minute, say.

<b>Hint 3: </b> How long will it take the water to flow out?

<b>Hint 4: </b> How do you find the rate of change of the volume?

Re: Rate of change problem: conical tank

If it flows at `f` liters per minute, then it will take `V/f` minutes to empty out.

Do we use `(dV)/(dt)` to find rate of change of volume?

Then what?

X

If it flows at `f` liters per minute, then it will take `V/f` minutes to empty out.

Do we use `(dV)/(dt)` to find rate of change of volume?

Then what?

Re: Rate of change problem: conical tank

Yes, `(dV)/(dt)` is correct.

On the right hand side of our volume equation, what is constant and what varies as the water empties out?

X

Yes, `(dV)/(dt)` is correct. 

On the right hand side of our volume equation, what is constant and what varies as the water empties out?

Re: Rate of change problem: conical tank

`pi` is constant, but `r` and `h` vary as time goes on.

I use product rule:

so `(dV)/(dt) = pi( r^2 (dh)/(dt) + (h) 2r (dr)/(dt))`

But I'm lost again.

X

`pi` is constant, but `r` and `h` vary as time goes on.

I use product rule:

so `(dV)/(dt) = pi( r^2 (dh)/(dt) + (h) 2r (dr)/(dt))`

But I'm lost again.

Re: Rate of change problem: conical tank

What you have is correct, but it's more complicated than it needs to be.

What's the relationship between `r` and `h`?

Can you simplify `V` now?

X

What you have is correct, but it's more complicated than it needs to be.

What's the relationship between `r` and `h`? 

Can you simplify `V` now?

Re: Rate of change problem: conical tank

Radius is 1.5 m and height is 5 m, so

`r/h = 1.5/5 = 0.3`

So `r = 0.3h`

So now

`V = (pi r^2 h)/3 ` `= (pi (0.3h)^2 h)/3 ` `= 0.03 pi h^3`

I see it now

`(dV)/(dt) = 3(0.03 pi h^2)(dh)/(dt)` `=0.09 pi h^2(dh)/(dt)`

Where do we use `f`?

X

Radius is 1.5 m and height is 5 m, so 

`r/h = 1.5/5 = 0.3`

So `r = 0.3h`

So now 

`V = (pi r^2 h)/3 ` `= (pi (0.3h)^2 h)/3 ` `= 0.03 pi h^3`

I see it now

`(dV)/(dt) = 3(0.03 pi h^2)(dh)/(dt)` `=0.09 pi h^2(dh)/(dt)`

Where do we use `f`?

Re: Rate of change problem: conical tank

Well, `f` is just the rate of flow, so it's equal to the change in volume of the water. So we have

`f=0.09 pi h^2(dh)/(dt)`

Can you get the expression of the rate of change of the height now?

X

Well, `f` is just the rate of flow, so it's equal to the change in volume of the water. So we have

`f=0.09 pi h^2(dh)/(dt)`

Can you get the expression of the rate of change of the height now?

Re: Rate of change problem: conical tank

So is it this?

`(dh)/(dt) = f/(0.09pih^2)`

X

So is it this?

`(dh)/(dt) = f/(0.09pih^2)`

Re: Rate of change problem: conical tank

Yes, you are correct. What are the units?

X

Yes, you are correct. What are the units?

Re: Rate of change problem: conical tank

The units will be m/min

X

The units will be m/min

Re: Rate of change problem: conical tank

Correct.

X

Correct.

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