Question

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 1

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.