# Growth and Decay question

• Jun 21st 2009, 09:38 AM
JTG2003
Growth and Decay question
There's this problem I have to do for calc, and I'm having a hard time getting it started.

The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours.

A) Find the initial population

We've only done problems where the initial population has been given to us. Could someone explain how to find it?

Thanks, Jeff
• Jun 21st 2009, 10:10 AM
Revenantus
Hi,

Since we're modeling the number of bacteria, $y$, at time, $t$, according to the law of exponential growth;

$y=Ae^{kt}$

Where $A$ is a constant representing the initial number of bacteria.

From the question;

$125=Ae^{2k}$

$350=Ae^{4k}$
• Jun 21st 2009, 10:12 AM
apcalculus
Quote:

Originally Posted by JTG2003
There's this problem I have to do for calc, and I'm having a hard time getting it started.

The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours.

A) Find the initial population

We've only done problems where the initial population has been given to us. Could someone explain how to find it?

Thanks, Jeff

The model fits the equation $y = A B^t$

where A= initial population, y is population at time t. We're given two conditions:

$125 = A B^2$ at t=2 hours
$350 = A B^4$ at t=4 hours

Divide side by side the second equation by the first:

$2.8 = B^2$
$B = 1.67332005$

Plug this value into one of the conditions above to find A.

Good luck!!
• Jun 21st 2009, 03:57 PM
JTG2003
Quote:

Originally Posted by apcalculus
The model fits the equation $y = A B^t$

where A= initial population, y is population at time t. We're given two conditions:

$125 = A B^2$ at t=2 hours
$350 = A B^4$ at t=4 hours

Divide side by side the second equation by the first:

$2.8 = B^2$
$B = 1.67332005$

Plug this value into one of the conditions above to find A.

Good luck!!

I appreciate the reply. I'm a little confused about what you mean by

"Divide side by side the second equation by the first:"

.. I just don't see what you did to get from those 2 equations to the next line
• Jun 21st 2009, 04:06 PM
apcalculus
Quote:

Originally Posted by JTG2003
I appreciate the reply. I'm a little confused about what you mean by

"Divide side by side the second equation by the first:"

.. I just don't see what you did to get from those 2 equations to the next line

350 by 125 gives 2.8
ab^4 by ab^2 gives b^2 since a (the initial population) is nonzero.