Question

A bacterial colony grows via cell division where each mother bacterium independently produces two daughter cells in 20 minutes. If the concentration of bacteria is \( 10^4 \, \text{cm}^{-3} \), the colony becomes harmful. Starting from a colony with an initial concentration of 5 cm$^{-3}$, the time taken (in minutes) for the colony to become harmful is ......... (Round off to nearest integer)
 

Hint:

Exponential growth is common in bacterial colonies, and the doubling time can be used to calculate the time for a population to reach a certain threshold.

Answer 1

Correct Answer: 210

Step 1: Understanding the Growth Model.
The bacterial colony grows exponentially, meaning that the number of bacteria doubles every fixed time period (20 minutes). We need to determine how many doubling times it takes for the initial concentration to increase from 5 cm$^{-3}$ to \( 10^4 \, \text{cm}^{-3} \), at which point the colony becomes harmful.

Step 2: Applying the Growth Formula.
The growth of the colony follows the formula: \[ N = N_0 \cdot 2^n \] Where: \( N \) is the final concentration, \( N_0 \) is the initial concentration, \( n \) is the number of doubling times. We can rearrange the formula to solve for \( n \): \[ n = \frac{\log(N / N_0)}{\log(2)} \] Substitute the given values: \( N_0 = 5 \, \text{cm}^{-3} \), \( N = 10^4 \, \text{cm}^{-3} \), and calculate \( n \). Then multiply by the doubling time (20 minutes) to find the total time.

Step 3: Conclusion.
The time taken for the colony to become harmful is 100 minutes.