A few questions ...

**finalfantasy** · November 27th 2007, 01:57 PM

Hello. I have a few questions I am having trouble with.

12. A bacterium has 25 cells, and in 10 hours grow to 350. What is the doubling period of the bacteria? Use the formula y=c(a)^t/d

13. In 1990, at t=0, the population of a city was 48 500. In the year 2000, its population was 50 000. If the population grows exponentially, find the groth rate using the formula P=k(a)^t

**earboth** · November 27th 2007, 09:01 PM

Originally Posted by finalfantasy

...14^\frac1{10}}

12. A bacterium has 25 cells, and in 10 hours grow to 350. What is the doubling period of the bacteria? Use the formula y=c(a)^t/d

...

Hello,

we have: $t_0 = 0,~ y_0 = 25$ [1]

we know: $t = 10,~ y_10 = 350$ [2]

Calculate the base first using $y = c \cdot a^t$

[1]: $25 = c \cdot a^0~\implies~ \boxed{c=25}$

[2] $350=25 \cdot a^10~\implies~ \boxed{a=14^\frac1{10}}$

If the amount of cells has doubled we have 50 cells. Use all known values to calculate the time:
$50=25 \cdot \left(14^\frac1{10} \right)^t~\implies~ 2=\left(14^\frac1{10} \right)^t$ $~\implies~\ln(2)=\frac1{10} \cdot \ln(14) \cdot t~\implies~ t=10 \cdot \frac{\ln(2)}{\ln(14)}\approx 2.626\ h\approx 2h\ 37\ min \ 35\ s$

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