Respuesta :
Answer:
Both cultures will have the same population after 11 hours.
Step-by-step explanation:
Exponential equation for population growth:
The exponential equation for population growth is given by:
[tex]P(t) = P(0)(1+r)^t[/tex]
In which P(0) is the initial population and r is the growth rate, as a decimal.
A culture of 100 bacteria grows at a continuous rate of 20%.
This means that [tex]P(0) = 100, r = 0.2[/tex]. So
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]P_{1}(t) = 100(1+0.2)^t[/tex]
[tex]P_{1}(t) = 100(1.2)^t[/tex]
The other culture starts to hours later, and we want to take that time as a parameter. So
[tex]P_{1}(2) = 100(1.2)^2 = 144[/tex]
Which means that when culture 2 growth starts, the population size of culture 1 is described by the following equation:
[tex]P_{1}(t) = 144(1.2)^t[/tex]
Two hours later 60 of the same type of bacteria are placed in a culture that allows a 30% growth rate.
This means that [tex]P(0) = 60, r = 0.3[/tex]. So
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]P_{2}(t) = 60(1+0.3)^t[/tex]
[tex]P_{2}(t) = 60(1.3)^t[/tex]
After how many hours do both cultures have the same population?
This is t for which:
[tex]P_{1}(t) = P_{2}(t)[/tex]
So
[tex]144(1.2)^t = 60(1.3)^t[/tex]
[tex]\frac{(1.3)^t}{(1.2)^t} = \frac{144}{60}[/tex]
[tex](\frac{1.3}{1.2})^t = \frac{144}{60}[/tex]
[tex]\log{(\frac{1.3}{1.2})^t} = \log{\frac{144}{60}}[/tex]
[tex]t\log{\frac{1.3}{1.2}} = \log{\frac{144}{60}}[/tex]
[tex]t = \frac{\log{\frac{144}{60}}}{\log{\frac{1.3}{1.2}}}[/tex]
[tex]t = 11[/tex]
Both cultures will have the same population after 11 hours.
