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Answer:
Explanation:
Let X be the number of bacteria at time t, and Xo be the initial number of bacteria
Bacteria grows exponentially, the exponential growth model is thus:
X = Xo
[<em>e</em> is the exponential sign]
k is the growth constant = growth rate
At t = 20 minutes, X = 100
At t = 40 minutes, X = 2000
Substituting that into the formula
(i) ... 100 = Xo
(ii) ... 2000 = Xo
<em>Divide (ii) by (i)</em>
20 =
<em>[Xo cancels Xo. Division of values raised to a power is done by subtracting their powers]</em>
<em>Take the natural log of both sides</em>
㏑20 = 20k
<em>[taking the ㏑ cancels the exponential]</em>
k =
<em>We can now substitute k to solve one of the equations</em>
<em>substituting k in (ii)</em>: 2000 = Xo
2000 = Xo
2000 = Xo
<em>Making Xo the subject of the formula</em>
Xo =
Xo is approximately 5 cells.
THE DOUBLING TIME
The doubling time is the time it takes for the population to double, so 5 cells become 10 cells
Since Xo = 5
Given that X = Xo
When X=10,
10 = 5
<em>solving for t:</em>
Take ln of both sides
0.15t = ln 2
t =
t =4.62 minutes
POPULATION AFTER 65 MINUTES
X = 5
X = 85770
WHEN WILL THE POPULATION REACH 12000
12000 = 5
=240
take ln of both sides
0.15t = ln 2400
t = 51.89 minutes (approximately 52 minutes)