A certain population of bacteria becomes (4÷3) times its initial popu

1.	A certain population of bacteria becomes (4÷3) times its initial population every day . A sample of 300 bacteria is taken for study . The expression becomes 300× (4÷3)^x models the number of bacteria in the sample after x days. Find the population (approx.)at the end of first day ,second day and third day . After which day will the population be more than three times of original ?
Answer» Given: The expression for POPULATION of bacteria after $x$ days is given by $\quad\quad N_{x}=300\times \left(\frac{4}{3}\right)^{x}$ To find: Population of bacteria at the end of first DAY second day third day and after which day, the population of bacteria will be more than three times of original. Solution: The expression for population of bacteria after $x$ days is given by $\quad N_{x}=300\times \left(\frac{4}{3}\right)^{x}$ Then the population of bacteria at the end of the first day is $\quad N_{1}=300\times \frac{4}{3}$ $\quad \boxed{\color{blue}{N_{1}=400}}$ The population of bacteria at the end of the second day is $\quad N_{2}=300\times \left(\frac{4}{3}\right)^{2}$ $\Rightarrow N_{2}=300\times \frac{16}{9}$ $\Rightarrow \boxed{\color{blue}{N_{2}\approx 533}}$ The population of bacteria at the end of the third day is $\quad N_{3}=300\times \left(\frac{4}{3}\right)^{3}$ $\Rightarrow N_{3}=300\times \frac{64}{27}$ $\Rightarrow \boxed{\color{blue}{N_{3}\approx 711}}$ Let, after $p-th$ day, the population of bacteria will be more than three times of original $\Rightarrow N_{p}>3\times 300$ $\Rightarrow 300\times\left(\frac{4}{3}\right)^{p}>3\times 300$ $\Rightarrow \left(\frac{4}{3}\right)^{p}>3$ $\Rightarrow p\:log\left(\frac{4}{3}\right)>log3$ $\Rightarrow p>\frac{log3}{\frac{4}{3}}$ $\Rightarrow p>3.82$ $\Rightarrow \color{blue}{p\approx 4}$ Therefore after $4$ days, the population of bacteria will be more than three times of original.