If are Given t(100%) as 45 minutes
Find k from the above data
Then find ratio

The ratio of population of bacteria at 30 minutes and 90 minutes to the initial population can be calculated by using the concept of bacterial growth and first-order kinetics. Let's break down the calculation and explanation into several sections.

Understanding Bacterial Growth:
Bacterial growth refers to the increase in the number of bacteria over time. It can be described using different growth models, and one of the commonly used models is exponential growth. In exponential growth, the number of bacteria doubles in a fixed time interval known as the generation time.

Generation Time:
The generation time is the time it takes for a population of bacteria to double in size. In this case, the given generation time is 45 minutes. This means that after every 45 minutes, the bacterial population will double.

First-Order Kinetics:
First-order kinetics describes a process in which the rate of reaction is directly proportional to the concentration of the reactant. In the case of bacterial growth, the rate of growth is directly proportional to the population size. This means that the rate at which bacteria multiply is dependent on the number of bacteria present at any given time.

Calculating the Ratio:
To calculate the ratio of the bacterial population at 30 minutes and 90 minutes to the initial population, we can use the concept of generation time and exponential growth.

1. Find the number of generations:
- 30 minutes / 45 minutes = 0.67 generations
- 90 minutes / 45 minutes = 2 generations

2. Calculate the ratio at 30 minutes:
- The population at 30 minutes is 2^0.67 times the initial population (since 0.67 generations have passed).
- Ratio at 30 minutes = 2^0.67

3. Calculate the ratio at 90 minutes:
- The population at 90 minutes is 2^2 times the initial population (since 2 generations have passed).
- Ratio at 90 minutes = 2^2

Final Answer:
The ratio of the population of bacteria at 30 minutes to the initial population is 2^0.67, and the ratio at 90 minutes is 2^2. These ratios represent the exponential growth of bacteria based on the given generation time and first-order kinetics.

Explanation:
The generation time of 45 minutes indicates that the bacterial population will double every 45 minutes. Therefore, after 30 minutes, the population will be less than double, but after 90 minutes, it will be more than double. The ratio at 30 minutes will be less than 2, while the ratio at 90 minutes will be greater than 2. This is because the population growth is exponential and accelerates with time.

By calculating the ratios, we can understand the relative increase in the bacterial population at different time points compared to the initial population. This information is valuable in various fields, such as microbiology, epidemiology, and environmental science, where understanding bacterial growth dynamics is crucial.