Question

If the generation time of a bacterium is 40 minutes and a culture containing $${10}^7$$ cells / ml is grown for 4 hours, then calculate its population after that period:
(a) $$64\times {10}^7$$
(b) $$32\times {10}^7$$
(c) $$6\times {10}^7$$
(d) $$40\times {10}^7$$

Answer 1

Hint: Convert hours into minutes by multiplying the given hour with 60. Use the information that a bacteria splits into two parts after 40 minutes and calculate the total number of bacteria formed after 4 hours. Calculate the value for 1 bacteria and multiply with $${10}^7$$ to get the correct option.


Complete step by step answer:
Here, we have to find the total population of bacteria after 4 hours if one bacteria splits into two parts after each 40 minutes. We have a total of $${10}^7$$ bacteria.
Let us calculate the total population formed from 1 bacteria.
We have 4 hours given.
We know that, 1 hour = 60 minutes.
$$\Rightarrow$$ 4 hours = $$4\times 60$$ = 240 minutes.
Now, the population of bacteria formed after 40 minutes = 2.
After 40 $$\times$$ 2 = 80 minutes, population = 2 $$\times$$ 2 = $$2^2$$.
After 40 $$\times$$ 3 = 120 minutes, population = $$2^2\times 2=2^3$$.
After 40 $$\times$$ 4 = 160 minutes, population = $$2^3\times 2=2^4$$.
After 40 $$\times$$ 5 = 200 minutes, population = $$2^4\times 2=2^5$$.
After 40 $$\times$$ 6 = 240 minutes, the population becomes = $$2^5\times 2=2^6$$.
So, after 240 minutes or 4 hours 1 bacteria will form $$2^6$$ = 64 bacteria. Since, we have $${10}^7$$ bacteria in total, so each one will split into 64 bacteria. Therefore, we have,
Total bacteria formed from the culture = $$64\times {10}^7$$.

So, the correct answer is “Option A”.

Note: One may note, we have converted 4 hours into 240 minutes. One can also apply the reverse process by converting 40 minutes into required hours by dividing it with 60 and then calculate the population after 4 hours. We have first calculated the population for 1 bacteria because if we will directly consider $${10}^7$$ bacteria then we may get confused.