Question

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 will be \( 4N \) in

• A. \( 2 \) hours
• B. \( 4 \) hours
• C. \( 6 \) hours
• D. \( 8 \) hours

Hint:

In exponential growth, if doubling takes time \( T \), then quadrupling takes \( 2T \).

Answer 1

The Correct Option is D

Step 1: Use the exponential growth model.
Since the rate of increase is proportional to the number present, \[ N(t) = N_0 e^{kt} \] Step 2: Use the doubling condition.
Given that the population doubles in \( 4 \) hours, \[ 2N = N e^{4k} \Rightarrow e^{4k} = 2 \] Step 3: Find the growth constant.
\[ k = \frac{\ln 2}{4} \] Step 4: Find time for population to become \( 4N \).
\[ 4N = N e^{kt} \Rightarrow e^{kt} = 4 \Rightarrow kt = \ln 4 = 2\ln 2 \] Step 5: Substitute the value of \( k \).
\[ t = \frac{2\ln 2}{\ln 2/4} = 8 \] Step 6: Conclusion.
The number of bacteria becomes \( 4N \) in \( 8 \) hours.