Question

The count of bacteria in a culture was initially 5,25,000. If it is increasing at the rate of \[3\dfrac{1}{2}\% \] per hour, find the count of bacteria at the end of 2 hours.

Answer 1

Hint: Here in this question, we have to find the count of bacteria which increase at the end of two hours. This question can be solve, by using a formula of compound interest i.e., \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\], where $A$ be total amount or count of bacteria, $P$ be the initially count of bacteria, $R$ be the rate of increase and $n$ be the time on substituting the values in formula and by further simplification we get the required solution.

Complete step by step solution:
Given, The initial count of the cultural bacteria is 5,25,000. Since the population of bacteria increases at the rate of \[3\dfrac{1}{2}\% \] per hour.
So, we have to calculate the total count of cultured bacteria at the end of 2 hour.
Now use the formula of compound interest,
\[ \Rightarrow \,\,A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\]
Where,
P is the initial count of bacteria i.e., \[P = 5,25,000\]
R is the increasing rate i.e., \[R = 3\dfrac{1}{2}\% = \dfrac{7}{2}\% \]
n is the time i.e., \[n = 2\]
A is the total count of bacteria at the end of time.
On substituting the values in formula, we have
\[ \Rightarrow \,\,A = 5,25,000{\left( {1 + \dfrac{{\left( {\dfrac{7}{2}} \right)}}{{100}}} \right)^2}\]
On simplification, then we have
\[ \Rightarrow \,\,A = 5,25,000{\left( {1 + \dfrac{7}{{2 \times 100}}} \right)^2}\]
\[ \Rightarrow \,\,A = 5,25,000{\left( {1 + \dfrac{7}{{200}}} \right)^2}\]
Take 200 as LCM, then
\[ \Rightarrow \,\,A = 5,25,000{\left( {\dfrac{{200 + 7}}{{200}}} \right)^2}\]
\[ \Rightarrow \,\,A = 5,25,000{\left( {\dfrac{{207}}{{200}}} \right)^2}\]
On simplification, we get
\[ \Rightarrow \,\,A = 5,25,000{\left( {1.035} \right)^2}\]
\[ \Rightarrow \,\,A = 5,25,000\left( {1.071225} \right)\]
On multiplication, we get
\[ \Rightarrow \,\,A = 5,62,393.125\]
\[ \Rightarrow \,\,A \approx 5,62,393\]

Therefore, At the end of 2 hour the approximate count of cultured bacteria is \[5,62,393\].

Note:
It can be seen that the compound interest formula is a very useful tool in calculating the future value of an investment, rate of investment, Increase or Decrease in Population etc. using the formula of compound interest i.e., \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\], we can find increasing or decreasing the initial amount or population at the different periods of time level on the terms of rate R.