# Thread: A few questions ...

1. ## A few questions ...

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

2. 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: $\displaystyle t_0 = 0,~ y_0 = 25$ [1]

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

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

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

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

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