Question

In a laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2.5% per hour. Find the bacteria at the end of two hours, if the count was initially 5,06,000.

Answer 1

Hint:- We had to only apply compound interest formula i.e. \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\] where A is the bacteria after two hours, P is the initial number of bacteria, R is the compounded rate on which bacteria is increasing and n is the number of hours after which we had to find the count of bacteria.

Complete step-by-step answer:

As we know bacteria are increasing per hour. So, we cannot apply a percentage formula to find the 2.5% of the given bacteria because it will give us the bacteria after one hour.
So, now we know that if we are some number or amount which is increasing with time and the rate at which is also given the we had to apply compound interest formula which states that if P will be the initial amount and R is the rate of changing the amount per hour/year (according to the condition in the question) and we had to find the amount after n hours/years (according to the condition in the question), then we apply the formula \[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}\] to find the amount after n hours/years (according to the condition in the question).
So, applying the formula. We get,
\[A = 506000{\left( {1 + \dfrac{{2.5}}{{100}}} \right)^2} = 506000{\left( {\dfrac{{100 + 2.5}}{{100}}} \right)^2} = 506000{\left( {\dfrac{{102.5}}{{100}}} \right)^2} = 506000\left( {\dfrac{{102.5}}{{100}}} \right)\left( {\dfrac{{102.5}}{{100}}} \right)\]
Now solving the above equation. We get,
\[A = 50.6 \times 102.5 \times 102.5 = 531616.25\]
Hence, the number of bacteria after two hours will be approximately equal to 5,31,616.

Note:- Whenever we come with this type of problem then there is also another method to find the bacteria after two hours, and that is by using percentage formula. First, we had to find the 2.5% of initial number of bacteria and then add that to get the number of bacteria initially to get the number of bacteria after one hour after that we had to find the 2.5% of bacteria after one hour and then add that with the count of bacteria after one hour to get the required number of bacteria after two hours. But this will not be the efficient way because if the number of hours increases then the solution will become complicated and large.