Copy - Bacteria Growth Logistic Growth Project

Bacteria growth Logistic growth project

by: jackie chau

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Part a)

Bacteria Growth

Bacteria growth Logistic growth project

At time t = 0 hours, a bacterial culture weighs 1 gram. Two hours later, the culture weighs 2 grams. The maximum weight of the culture is 10 grams.

Part b)

Part e)

by: jackie chau

Part d)

Part c)

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Compared to our exact answer provided by the logistic growth function, we can see that using Euler's method, we got an underestimate with a 5.22% percent error.

Euler's Method Approximations

Time (t) in hours

Weight (W) in grams

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Graph of

Time (t) in hours

Weight/hour (dP/dt) in grams/hour

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Now that we have found the differential equation, we can now use Euler's method to estimate the culture's weight at t=5, which is shown below.

Using the inital condition (2, 2) we can estimate the culture's weight at t=5.

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Part d)

Write a logistic differential equation that models the growth rate of the culture's weight. Then repeat part b) using Euler's Method with a step size of h = 1. Compare the approximation with the exact answers.

The general formula for a logistic differential equation is:

Since k = 0.0405 and A = 10 in this case, our differential equation is:

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Part b)

Find the culture's weight after 5 hours.

To solve for the culture's weight after 5 hours, we can use our logistic growth function to estimate the weight.

When evaluated with a calculator, we get:

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In this case, we would have to substitute 5 for P(t) and solve for t:

Part e)

At what time is the culture's weight increasing most rapidly?

For logistic growth functions, the point at which the y-variable (in our case, this is weight of the bacteria culture) is increasing the fastest is the point at which the y-variable is exactly half of the carrying capacity. So, we would have to find the time were the bacteria culture's weight is exactly 5 grams.

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Bacteria Logistic Growth

Time (t) in hours

Weight (W) in grams

We can solve for the constant k with the inital condition (2, 2):

Making our final function:

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We can solve for the constant C with the inital condition (0, 1):

After plugging in 9 in place of C, we get:

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Part a)

Write a logistic equation that models the weight of the baterial culture.

Here's the general formula for a logistic equation:

Where:

• A = carrying capacity
• C = any constant
• k = any constant

After plugging in all our known values, we get:

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Part c)

When will the culture's weight reach 8 grams?

To find when P(t) = 8, we need to substitute that into P(t) to solve for t.

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