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.

Accepted Solution

Answer:(1) Let, P represents the size of the bacteria culture in t hours,According to the question,[tex]\frac{dP}{dt}\propto P[/tex][tex]\frac{dP}{dt}=kP[/tex][tex]\frac{dP}{P}=kdt[/tex]By integrating,[tex]\ln P=kt + C[/tex][tex]P=e^{kt+C}[/tex][tex]P=e^{kt}.e^C[/tex][tex]P=P_0 e^{kt}[/tex] Where [tex]e^C=P_0[/tex],We have,at t = 0, P = 200,[tex]\implies 200=P_0 e^0\implies P_0 = 200[/tex]at t = 6, P = 1200[tex]1200=200 e^{6k}\implies k = 0.299[/tex]Hence, the required function would be,[tex]P=200 e^{0.299t}[/tex](2) if t = 7,The population would be,[tex]P=200 e^{0.299\times 7}=1621.84126567\approx 1622[/tex](3) If P = 1750,[tex]1750=200 e^{0.299t}\implies t = 7.254[/tex]Hence, it will take about 8 years for the population to reach 1750.