Respuesta :
Answer:
It takes 8.7 hours for there to be 3300 bacteria present.
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.
2400 bacteria are placed in a petri dish.
This means that [tex]P(0) = 2400[/tex]
So
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]P(t) = 2400(1+r)^t[/tex]
The conditions are such that the number of bacteria is able to double every 19 hours.
This means that [tex]P(19) = 2P(0)[/tex]. We use this to find 1 + r.
[tex]P(t) = P(0)(1+r)^t[/tex]
[tex]2P(0) = P(0)(1+r)^{19}[/tex]
[tex](1+r)^{19} = 2[/tex]
[tex]\sqrt[19]{(1+r)^{19}} = \sqrt[19]{2}[/tex]
[tex]1 + r = 2^{\frac{1}{19}}[/tex]
[tex]1 + r = 1.03715504445[/tex]
So
[tex]P(t) = 2400(1+r)^t[/tex]
[tex]P(t) = 2400(1.03715504445)^t[/tex]
How long would it be, to the nearest tenth of an hour, until there are 3300 bacteria present?
This is t for which [tex]P(t) = 3300[/tex]. So
[tex]P(t) = 2400(1.03715504445)^t[/tex]
[tex]2400(1.03715504445)^t = 3300[/tex]
[tex](1.03715504445)^t = \frac{3300}{2400}[/tex]
[tex]\log{(1.03715504445)^t} = \log{\frac{33}{24}}[/tex]
[tex]t\log{1.03715504445} = \log{\frac{33}{24}}[/tex]
[tex]t = \frac{\log{\frac{33}{24}}}{\log{1.03715504445}}[/tex]
[tex]t = 8.7[/tex]
It takes 8.7 hours for there to be 3300 bacteria present.
