Question

A bacteria culture starts with 200 bacteria and grows at a rate proportional to its size. After 6 hours there will be 1200 bacteria (1) Express the population after I hours as a function of t. population: p(tepe (1.066-21) (unction of t) (b) What will be the population after 7 hours? 348125.2 (c) How long will it take for the population to reach 1750 ? Note: You can earn partial credit on this problem.

Answer 1

Answer:

(1) Let, P represents the size of the bacteria culture in t hours,

According to the question,

$\frac{dP}{dt}\propto P$

$\frac{dP}{dt}=kP$

$\frac{dP}{P}=kdt$

By integrating,

$\ln P=kt + C$

$P=e^{kt+C}$

$P=e^{kt}.e^C$

$P=P_0 e^{kt}$ Where $e^C=P_0$,

We have,

at t = 0, P = 200,

$\implies 200=P_0 e^0\implies P_0 = 200$

at t = 6, P = 1200

$1200=200 e^{6k}\implies k = 0.299$

Hence, the required function would be,

$P=200 e^{0.299t}$

(2) if t = 7,

The population would be,

$P=200 e^{0.299\times 7}=1621.84126567\approx 1622$

(3) If P = 1750,

$1750=200 e^{0.299t}\implies t = 7.254$

Hence, it will take about 8 years for the population to reach 1750.