Question

The bacteria increases at the rate proportional to the number of bacteria present. If the original number $N$ doubles in $4$ hours, then the number of bacteria in $12$ hours will be

• A. $3N$
• B. $4N$
• C. $6N$
• D. $8N$

Hint:

In exponential growth, if a quantity doubles in time $T$, then in $3T$ it becomes eight times.

Answer 1

The Correct Option is D

Step 1: Writing the growth model.
Since the rate of increase is proportional to the number present, the growth follows exponential law: \[ N(t) = N_0 e^{kt} \]
Step 2: Using the given condition.
Given that the number doubles in $4$ hours: \[ 2N = N e^{4k} \Rightarrow e^{4k} = 2 \]
Step 3: Finding the population after $12$ hours.
\[ N(12) = N e^{12k} = N \left(e^{4k}\right)^3 = N(2)^3 = 8N \]
Step 4: Conclusion.
The number of bacteria after $12$ hours is $8N$.