a 1 ml sample of a bacterium culture at 37 celcius is taken and diluted to 10L.A 1 ml sample of the diluted culture is spead on a culture plte .10 minutes later another a ml sample is taken from the original culture diluted and spread inthe same way. the two plates are incubated for 24 hrs.the first sample exhibits 48 colonies and the second 72 colonies .if we assume that each colony originates with a single bacterium,what is the generation time(time required for doubling the population).

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To determine the generation time of the bacterium culture based on the information provided, we need to analyze the growth of the bacteria in both samples. The generation time is the time it takes for a population of bacteria to double in number. Here’s how we can calculate it step by step.

Understanding the Dilution and Growth Process

Initially, you have a 1 ml sample of a bacterium culture that is diluted to 10 liters. This means the original concentration of bacteria is significantly reduced. When you take a 1 ml sample from this diluted culture and spread it on a culture plate, you are effectively sampling a very small portion of the total bacteria present in the diluted solution.

Analyzing the First Sample

From the first dilution, you observed 48 colonies after incubation. Since each colony originates from a single bacterium, this means that the 1 ml sample taken from the diluted culture contained 48 bacteria. To find the total number of bacteria in the original 1 ml culture before dilution, we can use the dilution factor:

• Original culture volume = 1 ml
• Diluted culture volume = 10 L = 10,000 ml
• Dilution factor = 10,000 ml / 1 ml = 10,000
• Number of bacteria in the original culture = 48 colonies × 10,000 = 480,000 bacteria

Analyzing the Second Sample

For the second sample, you took another 1 ml from the same original culture after 10 minutes and spread it on a plate, which resulted in 72 colonies. Using the same dilution factor, we can calculate the number of bacteria in the original culture:

• Number of bacteria in the second sample = 72 colonies × 10,000 = 720,000 bacteria

Calculating the Growth Rate

Now that we have the number of bacteria from both samples, we can determine the growth that occurred in the 10 minutes between the two samples. The increase in the number of bacteria is:

• Increase = 720,000 - 480,000 = 240,000 bacteria

Next, we can calculate the growth rate using the formula for exponential growth:

N(t) = N0 × 2^(t/g)

Where:

• N(t) = the number of bacteria at time t
• N0 = the initial number of bacteria
• t = time elapsed (in minutes)
• g = generation time (in minutes)

For our case:

• N0 = 480,000
• N(t) = 720,000
• t = 10 minutes

Setting Up the Equation

We can rearrange the equation to solve for the generation time (g):

720,000 = 480,000 × 2^(10/g)

Dividing both sides by 480,000 gives:

1.5 = 2^(10/g)

Taking Logarithms

To solve for g, we can take the logarithm of both sides:

log(1.5) = (10/g) × log(2)

Now, rearranging gives:

g = 10 × log(2) / log(1.5)

Calculating the Values

Using a calculator:

• log(2) ≈ 0.3010
• log(1.5) ≈ 0.1761

Substituting these values into the equation:

g = 10 × 0.3010 / 0.1761 ≈ 17.06 minutes

Final Thoughts

The generation time for the bacterium culture, based on the data provided, is approximately 17.06 minutes. This means that under the conditions of your experiment, the population of bacteria doubles roughly every 17 minutes. Understanding this growth rate is crucial for applications in microbiology, such as optimizing culture conditions or predicting bacterial behavior in various environments.