Respuesta :
Answer:
[tex]r=1.57[/tex], [tex]P(t)=71e^{3t\ln2}[/tex], [tex]P(8)=1191182336[/tex],[tex]P'(8)=2476994033[/tex] [tex]t=2.713\; hr[/tex]
Step-by-step explanation:
[tex](a)[/tex] Find the relative growth rate.
[tex]P=Ae^{rt}[/tex] where given [tex]A=71, P(t)=120\;\;and \;\; t=\frac{1}{3}[/tex]
Find the value of [tex]r[/tex]
[tex]\Rightarrow 120=71 e^{\frac{r}{3} }[/tex]
[tex]\Rightarrow \frac{120}{71} =e^{\frac{r}{3} }[/tex]
[tex]\Rightarrow 1.69=e^{\frac{r}{3} }[/tex]
[tex]\Rightarrow \ln1.69=\frac{r}{3}[/tex]
[tex]\Rightarrow 3\ln1.69=r \;\;\; or \;\;\; r=1.57[/tex]
[tex](b)[/tex] Find an expression for the number of cells after t hours.
[tex]P(t)=71e^{3t\ln2}[/tex]
[tex](c)[/tex] Find the number of cells after 8 hours.
[tex]P(8)=71e^{3\times 8\times \ln2}[/tex]
[tex]\Rightarrow P(8)=1191182336[/tex]
[tex](d)[/tex] Find the rate of growth after 8 hours.
[tex]P(t)=71e^{3\times t\times \ln2}[/tex]
Now differentiate w.r.t. [tex]t[/tex]
[tex]\Rightarrow P'(t)=71\times 3\ln2\times e^{3t\ln2}[/tex]
[tex]\Rightarrow P'(8)=3573547008\ln2[/tex]
[tex]=2476994033[/tex]
[tex](e)[/tex] When will the population reach 20,000 cells?
[tex]20,000=71e^{3t\ln2}[/tex]
[tex]\Rightarrow \ln\frac{20000}{71} =3t\ln2[/tex]
[tex]\Rightarrow \frac{ \ln\frac{20000}{71}}{3\ln2} =t[/tex]
S0, [tex]t=2.713 hr[/tex]
