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. \( 4N \)
• B. \( 3N \)
• C. \( 8N \)
• D. \( 6N \)

Hint:

For exponential growth problems, use the formula \( N(t) = N_0 e^{kt} \) and solve for the rate constant using known values.

Answer 1

The Correct Option is C

Step 1: Understanding the exponential growth.
The problem describes exponential growth, which follows the equation: \[ N(t) = N_0 e^{kt} \] where \( N_0 \) is the initial number of bacteria, \( k \) is the rate constant, and \( t \) is the time.

Step 2: Using the doubling time.
Since the number of bacteria doubles in 4 hours, we have the relation: \[ 2N_0 = N_0 e^{4k} \] Solving for \( k \), we get \( e^{4k} = 2 \), so \( k = \frac{\ln 2}{4} \).

Step 3: Finding the number of bacteria after 12 hours.
Using the equation \( N(t) = N_0 e^{kt} \), we calculate the number of bacteria after 12 hours: \[ N(12) = N_0 e^{12k} = N_0 e^{12 \cdot \frac{\ln 2}{4}} = N_0 \cdot 8 \] Thus, the number of bacteria after 12 hours is \( 8N \), which makes option (C) the correct answer.