The count in a bacteria culture was 900 after 20 minutes and 1400 after 40 minutes. Assuming the count obeys a continuous exponential growth pattern. What was the initial size of the culture?

Question

Find the doubling period.

Find the population after 115 minutes.

When will the population reach 10000?

Answer 1

Exponential function for geometric growth/decay

A(t)=A₀b^t

A₀ is initial amount,,when t=0

b is base for growth/decay ,,,b>1 growth and b<1 decay

t is time unit,,for this problem t unit is hour

14/9=(A₀b^(2/3))/(A₀b^(1/3))

14/9=b^1/3

b=(14/9)³=3.76... meaning growth rate is 376% per hour

900=A₀((14/9)³)^1/3

A₀=9*900/14=578

the amount doubles when A(t2)/A(t1)=2

2=A₀3.76^t2/A₀3.67^t1

2=3.76^(t2-t1)

ln(2)=(t2-t1)ln(3.76)

t2-t1=doubling time)=ln(2)/ln(3.76)=.52 hour

A(115 min)=A(23/12)=578(3.76^(23/12))

10000=578(3.76^t)

t=ln(10000/578)/n(3.76)