Exponential/Log Function help

**Hockey_Guy14** · January 23rd 2008, 06:10 AM

I Can't seem to figure out how to show the work for this question

For a biology experiment, there are 50 cells present. After 2 hours there are 1600 bacteria. How many bacteria would there be after 6 hours?

The answer is 1 638 400 but I don't know how to get to that answer. Can someone help me out?

**earboth** · January 23rd 2008, 07:33 AM

Originally Posted by Hockey_Guy14

I Can't seem to figure out how to show the work for this question

For a biology experiment, there are 50 cells present. After 2 hours there are 1600 bacteria. How many bacteria would there be after 6 hours?

The answer is 1 638 400 but I don't know how to get to that answer. Can someone help me out?

Hello,

we assume exponential growth.
The amount of bacteria with respect to time can be calculated by:

$a(t) = a(0) \cdot e^{k\cdot t}$ .......... Thus you have to calculate the constants a(0) and k from the given conditions:

t = 0 and a(0) = 50

t = 2 and a(2) = 1600. Therefore:

$1600 = 50 \cdot e^{k \cdot 2}$

$32 = e^{k \cdot 2}$

$\ln(32) = k \cdot 2 ~\implies~ k = \frac12 \cdot \ln(32)$

Now you can calculate a(6):

$a(6)= 50 \cdot e^{\frac12 \cdot \ln(32) \cdot 6}=50 \cdot 32^3= 1,638,400$

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