Question

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

Answer 1

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`

Answer 2

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`