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•  » [CALCULUS] Differentiation and rate of changes PROBLEM

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## Topic review (newest first)

bob bundy
2012-11-09 00:07:22

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

Anonymous
2012-11-08 20:16:04

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

bob bundy
2012-11-01 22:00:45

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

anonimnystefy
2012-11-01 20:13:55

#### 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?

bob bundy
2012-11-01 19:25:53

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

Kevin123123
2012-10-31 12:17:06

#### 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

bob bundy
2012-10-31 11:25:51

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

anonimnystefy
2012-10-31 09:46:35

And c=x0.

bob bundy
2012-10-31 09:26:42

This is what I've got so far.

therefore

So

We want t when x = 0

Bob

anonimnystefy
2012-10-31 09:06:43

#### 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.

bob bundy
2012-10-31 09:04:09

hi Stefy,

So you do know the first and second bits ?

Bob

anonimnystefy
2012-10-31 09:01:51

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

bob bundy
2012-10-31 09:00:03

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

Kevin123123
2012-10-31 08:37:43

#### 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:

bob bundy
2012-10-31 08:26:30

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