Exponential Growth & Decay

**realintegerz** · January 11th 2009, 01:18 PM

Hi, I'm reviewing for a test that will happen this week and I need some help on these problems..

1) A yeast colony is growing exponentially, it starts with 300 individuals and 4 hours later it has 5000 individuals.

A) How long will it take to have 20,000 individuals?
b) What is the doubling time for this colony?

2) A radioactive substance has a half life of 8,000 years, how long does it take to lose 1% of the substance to radioactive decay?

In these problems, I'm just not getting how to figure out the time with the initial amount known and multiplying factor known... I tried doing some calculating but they aren't coming out right

I can do problems like these where the time is given but I have not been taught to find the time with only other info given

Any help would be appreciated

**skeeter** · January 11th 2009, 02:04 PM

basic equation ...

$y = y_0 e^{kt}$ , where $y_0$ is the initial population

$y = 300e^{kt}$

at $t = 4$ , $y = 5000$

$5000 = 300e^{4k}$

solve for $k$ ...

$\frac{3}{50} = e^{4k}$

$\ln\left(\frac{3}{50}\right) = 4k$

$k = \frac{1}{4}\ln\left(\frac{3}{50}\right) \approx -0.70335...$

if you have a store capability in your calculator, store this value in $k$ .

$20000 = 300e^{kt}$

$\frac{3}{200} = e^{kt}$

$\ln\left(\frac{3}{200}\right) = kt$

$t = \frac{1}{k}\ln\left(\frac{3}{200}\right) \approx 5.971$ hrs

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