# A bacteria culture doubles in size every 20 min. How long will it take for a culture of 50 bacteria to grow to a population of 250 000?Exponential Equations

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Given the initialpopulation is 50 and the growth rate is 2: find how long until the population is 250,000.

The formula is `A=P*b^t` where A is the population at time t, P the initial population and b the growth rate.

If t is in hours:

`250000=50*2^(3t)`

`==> 2^(3t)=5000`

`==>ln2^(3t)=ln5000`

`==>3tln2=ln5000`

`==>3t=(ln5000)/(ln2)`

`==>t=(ln5000)/(ln2)-:3~~4.095`

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**It will take approximately 4hrs 6 min for the population to reach 250000.**

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If you prefer to work in minutes:

`250000=50*2^(t/20)` where t is in minutes. Then:

`2^(t/20)=5000`

`t=(ln5000)/(ln2)*20~~245.754` minutes which is about 4.1 hours as above.

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