The count in a bacteria culture was 700 after 20 minutes and 1000 after 40 minutes. What was the initial size of the culture?

2 Answers
Jun 16, 2017

490 microorganisms.

Explanation:

I will assume exponential growth for bacteria. This means that we can model the growth with an exponential function:

#f(t)=A_0e^(kt)#

where #k# is the growth constant and #A_0# is the initial amount of bacteria.

Sub the two known values into the function to get two equations:

#700=A_0e^(20k)# (1)

#1000=A_0e^40k# (2)

Divide (2) by (1) to find #k#:

#1000/700=(cancel(A_0)e^(40k))/(cancel(A_0)e^(20k))#

#10/7=e^(40k-20k)=e^(20k)#

Take the natural log of both sides to isolate #k#:

#ln(10/7)=cancel(ln)cancel(e)^(20k)#

#ln(10/7)=20k#

#k=ln(10/7)/20#

Now that we have the growth constant, #k#, we can substitute one of the points in to solve for the initial amount, #A_0#:

#(40,1000)#

#1000=A_0e^(ln(10/7)/20*40)#

#A_0=1000/e^(0.0178*40)=490#

Jun 16, 2017

Initial culture size was #490#

Explanation:

The growth can be considered as a geometric progression with the same rate of growth after each interval of #20# minutes.

The rate of growth can be determined by #1000/700 =10/7#

In terms of the size of the initial population #(x)#

This means:

#x xx 10/7 rarr 700 xx 10/7 rarr 1000#
#0 " mins"color(white)(xxx) 20 " mins"color(white)(xxx) 40 " mins"#

So if we reverse the process we just divide by #10/7#

#x larr 10/7 div 700 larr 10/7 div larr 1000#

Remember that #div 10/7 = xx 7/10#

#1000 xx 7/10 =700#

#700 xx 7/10 = 490#