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•  » [CALCULUS] Differentiation and rate of changes PROBLEM

## #1 2012-10-31 07:05:04

Kevin123123
Guest

### [CALCULUS] Differentiation and rate of changes PROBLEM

A block of ice, in the shape of a rectangular prism, has the dimensions give below:
Width: x
Length: 3.75x
Height: 0.25x
You are to determine the time it takes for the block of ice to completely melt given the following information
1)    The block of ice retains its rectangular prism shape throughout the melting process.
2)    The rate of change of volume with respect to time is directly proportional to the surface area of the block
3)    After one time interval, the volume of the block of ice is a fraction of its initial volume. This fraction is given by the ratio of the area of the smallest face of the block to the total surface area of the block. The unit of time has not been specified, since the time taken for the block of ice to completely melt depends on the initial volume of the block of ice.

## #2 2012-10-31 07:16:19

Kevin123123
Guest

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Is it impossible to calculate if we dont know the width of the rectangular ice prism?

## #3 2012-10-31 08:07:41

Kevin123123
Guest

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

#### bob bundy wrote:

hi Kevin123123

Welcome to the forum.

I've had two goes at this and realised I'm missing something.  So I've deleted my attempts and I'll do it properly (I hope) and then post again.

Bob

alright, take ur time! thanks a lot!!

## #4 2012-10-31 08:18:44

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Thanks for saying that.  I'm having lots of trouble with this.

You can write an expression for dV/dt in terms of x you can do this I hope.

To solve this you need a way of changing x into t.

You know the initial volume in terms of x.

You know the ratio of volumes in terms of a ratio of surface areas. and do this

But is this a ratio that is fixed for all time intervals or just the first?

I think the problem is unsolvable unless it's for all time intervals.

Then you have dV in terms of dt and so maybe have enough to sort it all out.

Still thinking .............................

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #5 2012-10-31 08:26:30

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Cannot be for all time intervals or the amount of melting gets less and less and takes an infinite amount of time. ?????

What calculus are you expected to be using ?

I'll assume you have basic integration.

What about differential equations with separation of variables ?

And changing the variable to integrate with respect to.

Have you done numerical methods of integration ?

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #6 2012-10-31 08:37:43

Kevin123123
Guest

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

#### bob bundy wrote:

Cannot be for all time intervals or the amount of melting gets less and less and takes an infinite amount of time. ?????

What calculus are you expected to be using ?

I'll assume you have basic integration.

What about differential equations with separation of variables ?

And changing the variable to integrate with respect to.

Have you done numerical methods of integration ?

Bob

i can only use differentiation in this assignment!! So hard:mad:

## #7 2012-10-31 09:00:03

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Only differentiation.  eeekkkk!

Ok. Here's what I'd do.  I'm not at all confident but there's nothing else I can suggest.

Let's assume the sides of the block are always given by those expressions in x whatever size x is.

Then you can write

hence work out dV/dx using differentiation.

You can also write an expression for the surface area in terms of x and hence write the rate of change of volume

You should be able to work out both numbers.

Stick a minus on this as the volume is decreasing.

As x squared occurs in both of these you can get

This will integrate to an expression like this

m will be negative.  Note x will be the vertical axis and t across.

Use the initial volume and volume after one unit ot time to determine this gradient m.

When does the line cut  x = 0 ?

That's my best shot I'm afraid.  Maybe someone else on the forum will come in with a better suggestion.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #8 2012-10-31 09:01:51

anonimnystefy
Real Member

Offline

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

I do not understand what the 3rd piece of information is.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #9 2012-10-31 09:04:09

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

hi Stefy,

So you do know the first and second bits ?

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #10 2012-10-31 09:06:43

anonimnystefy
Real Member

Offline

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

#### Kevin123123 wrote:

A block of ice, in the shape of a rectangular prism, has the dimensions give below:
Width: x
Length: 3.75x
Height: 0.25x
You are to determine the time it takes for the block of ice to completely melt given the following information
1)    The block of ice retains its rectangular prism shape throughout the melting process.
2)    The rate of change of volume with respect to time is directly proportional to the surface area of the block
3)    After one time interval, the volume of the block of ice is a fraction of its initial volume. This fraction is given by the ratio of the area of the smallest face of the block to the total surface area of the block. The unit of time has not been specified, since the time taken for the block of ice to completely melt depends on the initial volume of the block of ice.

3) is bothering me. From iit I gather that (V+dV)/V is equal to 1/39.5 in which case the block of ice is melting very rapidly.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #11 2012-10-31 09:26:42

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

This is what I've got so far.

therefore

So

We want t when x = 0

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #12 2012-10-31 09:46:35

anonimnystefy
Real Member

Offline

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

And c=x0.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #13 2012-10-31 11:25:51

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

If the width decreases linearly then the volume will decrease by the cube of the value of 'm'

But we know that

So maybe

Does that help ?

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #14 2012-10-31 12:17:06

Kevin123123
Guest

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

#### bob bundy wrote:

If the width decreases linearly then the volume will decrease by the cube of the value of 'm'

But we know that

So maybe

Does that help ?

Bob

thanks, they are really helpful

## #15 2012-11-01 19:25:53

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Hi Kevin123123,

You're welcome.

I did write this reply a couple of days ago but then my laptop started playing up and, it seems, it never got on the site.

If you ever get an 'official' answer, I'd be pleased to know what it is.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #16 2012-11-01 20:13:55

anonimnystefy
Real Member

Offline

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

#### bob bundy wrote:

I did write this reply a couple of days ago but then my laptop started playing up and, it seems, it never got on the site.

Again?

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

## #17 2012-11-01 22:00:45

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Yes, it forgot it had a wireless card.

After a lot of fiddling, I tried the standard solution.  I turned it off and  on again, and it was normal.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

## #18 2012-11-08 20:16:04

Anonymous
Guest

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

Hi,
I have a task similar and I was wondering how you got dV/dt from dV/dx and what k is..

## #19 2012-11-09 00:07:22

bob bundy
Moderator

Online

### Re: [CALCULUS] Differentiation and rate of changes PROBLEM

hi Anonymous

Welcome to the forum.

It's an application of the chain rule:

Subject to the usual rules about differentiability you can do this with any three related variables.

Bob

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
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