Tissue Culture

Reuel · 2011-01-25 11:05:48

"Let A(t) represent the area of a tissue culture at time t and let B be the final area of the culture when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to the root of A(t). A reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to the root of A(t) and B - A(t). Formulate a differential equation and use it to show that the tissue grows fastest when t = B/3. Solve the differential equation for A(t). (Assume A(0) = 0.)"

In solving this problem I so far have

Separating produces

And solving this in a computer program yields a horrific answer and I don't think a computer is supposed to be used in the first place, though I suppose it is okay.

Do I have this set up correctly?

Thanks!

bobbym · 2011-01-25 11:51:47

Hi;

I like the setup as far as I know. But you are not integrating the LHS correctly.

I don't think a computer is supposed to be used in the first place, though

Not exactly right. For years students and mathematicians alike were using tables of integrals such as Pierce's. Before there was a Mathematica. In many instances it is just as strong. So if they had that advantage why can we not? When you are learning the methods of integration then you must do that by hand. When it is merely part of a larger problem then you can use tables or programs.

I checked the integral with both M's and a table of integrals.

Reuel · 2011-01-25 13:30:55

Hey...

Well, I hadn't integrated yet. I separated the variables and had the equality

and you solved the left side... and it's the same answer I was getting out of Maple, so thanks for verifying it!

Now this needs solved for A(t):

I assume it should just be solved algebraically. Here is my effort:

And so that is A(t). That is so complicated, though. Am I even moving in the right direction? What is the question really asking in relation to this?

Thanks for always hanging around to answer questions.

bobbym · 2011-01-25 13:34:33

Hi;

Hold on a second. I agree that is a tough one and you have to find the maxima of that yet.

I was wondering right here:

root of A(t) and B - A(t)

Did they mean the root of both of them because that simplifies the problem. What do you think?

Reuel · 2011-01-25 13:47:30

No, that's just the way I typed it. The problem says jointly proportional to

.

So the way it is typed in my work is correct to the problem. Good try though, that would have been better.

Hmmm...

bobbym · 2011-01-25 13:54:38

Hi;

Step 3 should look like this:

Reuel · 2011-01-25 13:58:54

I did not think it mattered since C is whatever it ends up being in the long run.

Is that better? I just skipped to the end.

bobbym · 2011-01-25 14:29:29

Hi;

When we solve for the constant C by using the initial conditions A(0) = 0

I am getting C = 0 so we are left with.

Reuel · 2011-01-26 00:20:52

So is B/3 supposed to be used in the differential equation, in the function A(t), or in a new derivative formed from the particular solution? It says to show that t = B/3 is where change occurs the quickest so it must have something to do with one of the two derivatives - the original or the derivative of A(t) as we have it.

Last edited by Reuel (2011-01-26 00:21:12)

bobbym · 2011-01-26 00:39:30

Hi Reuel;

I am not getting B / 3 when I am using either the answer differentiated and set to 0 or the original DE set to 0.

Reuel · 2011-01-26 12:19:53

Okay... so... I guess the key is to integrate by partial fractions because it gives a friendler form for answering the question.

You would have

Let

I do not remember how to integrate by partial fractions though. What are the steps for integrating the left side using partial fractions?

bobbym · 2011-01-26 12:34:23

Hi Reuel;

I use a simultaneous set of equations to do that. To get an idea use the parfrac switch in convert.

Reuel · 2011-01-26 12:48:54

I don't know what you mean.

Here is my attempt after looking in my old calc book.

And so my system of equations would be...

and

correct? Does this system solve for a = b? Is that how it should be integrated?

bobbym · 2011-01-26 12:56:58

Hi;

Okay... so... I guess the key is to integrate by partial fractions because it gives a friendler form for answering the question.

I do not think that will help. I will help you with it if you want but I think we solved the DE correctly. Another form can be done by algebraically manipulating the answer. We still will not get B / 3 as the maximum.

To answer the other problem you can get the partial fraction using Maple like this:

convert( rational function goes here , parfrac, variable you want )

Reuel · 2011-01-26 13:08:04

I was trying to solve the second half of the problem first. Here is the solution to the first half of the problem. Differentiate the differential equation and set that derivative equal to 0 to show that the growth of the culture is fastest when one-third of the whole possible surface is populated by the culture.

That is where the growth is at a maximum.

Last edited by Reuel (2011-01-26 13:12:03)

Reuel · 2011-01-26 13:10:41

The second half of the problem says to solve the differential equation for A(t). I want to do it by hand to see how to do it. We don't get to use computers on exams.

bobbym · 2011-01-26 13:12:46

Excellent! Wunderbar!

I tried that twice last night and I did not get that, I am sorry. I must have made an algebraic mistake along the way.

I think you are done. We are back to post #1. The integration of both sides.

Reuel · 2011-01-26 13:19:32

Thanks. I still need to figure out the second half of the problem by using integration by partial fractions by hand. Let me know if you come up with anything. I will work on it too.

bobbym · 2011-01-26 13:23:17

Did you see how to do it with maple, so you can at least check?

Reuel · 2011-01-26 13:30:50

I got an error. I am pretty tired... what did I type wrong? (See image.)

bobbym · 2011-01-26 13:35:38

Hi;

You left out the second parameter, the word parfrac.

Reuel · 2011-01-26 13:37:52

Great! I'll be back in however long with progress.

bobbym · 2011-01-26 13:40:37

Remember to have a multiplication sign between those two parentheses. Maple is particular in this case!

Try to get the partials by yourself. Let me know if you cannot.

Reuel · 2011-01-26 15:34:46

I solved it but it took several pages and I do not want to post it. I just hope it is correct.

bobbym · 2011-01-26 15:45:27

Hi Reuel;

Very good! You are the ideal poster, you do all the work! Make it easy for me!

Did you check it up against Maple?

Math Is Fun Forum

#1 2011-01-25 11:05:48

Tissue Culture

#2 2011-01-25 11:51:47

Re: Tissue Culture

#3 2011-01-25 13:30:55

Re: Tissue Culture

#4 2011-01-25 13:34:33

Re: Tissue Culture

#5 2011-01-25 13:47:30

Re: Tissue Culture

#6 2011-01-25 13:54:38

Re: Tissue Culture

#7 2011-01-25 13:58:54

Re: Tissue Culture

#8 2011-01-25 14:29:29

Re: Tissue Culture

#9 2011-01-26 00:20:52

Re: Tissue Culture

#10 2011-01-26 00:39:30

Re: Tissue Culture

#11 2011-01-26 12:19:53

Re: Tissue Culture

#12 2011-01-26 12:34:23

Re: Tissue Culture

#13 2011-01-26 12:48:54

Re: Tissue Culture

#14 2011-01-26 12:56:58

Re: Tissue Culture

#15 2011-01-26 13:08:04

Re: Tissue Culture

#16 2011-01-26 13:10:41

Re: Tissue Culture

#17 2011-01-26 13:12:46

Re: Tissue Culture

#18 2011-01-26 13:19:32

Re: Tissue Culture

#19 2011-01-26 13:23:17

Re: Tissue Culture

#20 2011-01-26 13:30:50

Re: Tissue Culture

#21 2011-01-26 13:35:38

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#22 2011-01-26 13:37:52

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#23 2011-01-26 13:40:37

Re: Tissue Culture

#24 2011-01-26 15:34:46

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#25 2011-01-26 15:45:27

Re: Tissue Culture

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