Respuesta :
Given:
A bacteria culture is started with 200 bacteria.
After 4 hours, the population has grown to 769 bacteria.
the population grows exponentially according to the formula:
[tex]P(t)=P_0(1+r)^t[/tex](a) Find the growth rate.
so, when:
[tex]\begin{gathered} P_0=200,t=4,P(t)=769 \\ 769=200(1+r)^4 \end{gathered}[/tex]Solve for r:
[tex]\begin{gathered} \frac{769}{200}=(1+r)^4 \\ 3.845=(1+r)^4 \\ \sqrt[4]{3.845}=1+r \\ 1+r=1.4 \\ r=1.4-1=0.4 \end{gathered}[/tex]So, the value of r = 0.4 = 40%
The growth rate = r = 40%
(b) If this trend continues, how many bacteria will there be in one day?
For one day, t = 24 hours
so,
[tex]\begin{gathered} P_t=200\cdot(1+0.4)^{24} \\ P_t=200\cdot1.4^{24}=642,840 \end{gathered}[/tex]Bacteria = 642,840
(c) How long will it take for this culture to triple in size?
So,
[tex]\begin{gathered} P_t=3\cdot200=600 \\ 600=200\cdot(1+0.4)^t \end{gathered}[/tex]Solve for t:
[tex]\begin{gathered} \frac{600}{200}=1.4^t \\ 3=1.4^t \\ \ln 3=t\cdot\ln 1.4 \\ t=\frac{\ln 3}{\ln 1.4}\approx3.265 \end{gathered}[/tex]Round your answer to the nearest tenth of an hour.
So, t = 3.3 hours
