Population growth

smithcity · 13-08-2009, 16:01

Hi all, I'm studying for a repeat exam and there's a question here that has me stumped.
I'm not going to ask you to solve it for me, but if you could point me in the right direction it would be appreciated.

At a time t = 0 a population of 1000 bacteria is introduced into a nutrient medium. At time t the size of the population is given by:

p = 9000(4 + t)

4(9 + t^2)

where time is t measured in hours. Find the maximum size of the population.

Any and all help would be greatly appreciated.

**-JammyDodger-** · 13-08-2009, 16:05

Find $\displaystyle\frac{dP}{dt}$ , and let it equal to 0. What do you know about letting a differentiated function equal 0?

smithcity · 13-08-2009, 16:18

I'm dimly aware that I need to differentiate the equation, let it = 0, an find two values for t. Determine which is min, and which is max. Then substitute it for t in the equation.

Is that the right track?

Just not too sure about how to differentiate this function
should I be using the :
dy = u.dv + v.du
dx dx dx
method?

**-JammyDodger-** · 13-08-2009, 16:22

Quote:

Originally Posted by smithcity

I'm dimly aware that I need to differentiate the equation, let it = 0, an find two values for t. Determine which is min, and which is max. Then substitute it for t in the equation.

Is that the right track?

That's exactly right.

Quote:

Just not too sure about how to differentiate this function
should I be using the :
dy = u.dv + v.du
dx dx dx

method?

That's the product rule.

Since the function is a fraction (quotient), you've to use the quotient rule. The notation in that wiki article is the hardest part to understand, really.

smithcity · 13-08-2009, 16:24

Yup sorry, was just coming back in to edit my mistake when you posted.

Ok, thanks for the help. I'll give it a lash like that.

13-08-2009, 16:01	#1
smithcity Registered User Join Date: Jul 2009 Location: Offaly Posts: 196 Adverts \| Friends	Population growth Hi all, I'm studying for a repeat exam and there's a question here that has me stumped. I'm not going to ask you to solve it for me, but if you could point me in the right direction it would be appreciated. At a time t = 0 a population of 1000 bacteria is introduced into a nutrient medium. At time t the size of the population is given by: p = 9000(4 + t) 4(9 + t^2) where time is t measured in hours. Find the maximum size of the population. Any and all help would be greatly appreciated. __________________ If you walk without rhythm, you wont attract the worm.

13-08-2009, 16:05	#2
-JammyDodger- Moderator Join Date: May 2008 Location: Galway/Dublin Posts: 2,676 Adverts \| Friends Mod: Anthropology, Sociology & Culture, Conspiracy Theories, Expand Your Horizons	Find $\displaystyle\frac{dP}{dt}$ , and let it equal to 0. What do you know about letting a differentiated function equal 0? __________________ Previous: Walden, by Henry David Thoreau Currenly: Hamlet's Mill, by Giorgio de Santillana and Hertha von Dechend Next: Travels in the Interior of Africa, by Mungo Park

13-08-2009, 16:18	#3
smithcity Registered User Join Date: Jul 2009 Location: Offaly Posts: 196 Adverts \| Friends	I'm dimly aware that I need to differentiate the equation, let it = 0, an find two values for t. Determine which is min, and which is max. Then substitute it for t in the equation. Is that the right track? Just not too sure about how to differentiate this function should I be using the : dy = u.dv + v.du dx dx dx method? __________________ If you walk without rhythm, you wont attract the worm. Last edited by smithcity; 13-08-2009 at 16:22.

13-08-2009, 16:24	#5
smithcity Registered User Join Date: Jul 2009 Location: Offaly Posts: 196 Adverts \| Friends	Yup sorry, was just coming back in to edit my mistake when you posted. Ok, thanks for the help. I'll give it a lash like that. __________________ If you walk without rhythm, you wont attract the worm.

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