Logistic Growth - Bacteria Project

Aaron Gonzalez P3

Logistic Growth - Bacteria

Worked Problems

Graphs

Bacteria Growth

Part A) Logistic Equation

Graph of f(t)

Part B) 5 Hours

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

Part C) 8 Grams

Part D) Euler's Method

Graph of f'(t)

Part E) Rate of Change

To find when the culture's weight will reach 8 grams we have to set the logistic equation equal to 8 and solve for t. After a bit of algebra, we find that the culture will weigh 8 grams after 8.848 hours.

Part C) When will the culture's weight reach 8 grams

Step 1: Set the logistic equation equal to 8 grams

Step 2: Rearrange and solve for t

To find when the culture's weight is increasing the fastest we have to find the maximum value of its rate of change. We can do this by finding the second derivative of the logistic equation and setting it equal to zero. This will give us the weight when its increasing the fastest and from there we can work backwards to find the time. In this case, we find that at time t = 5.425 hours the culture's weight is increasing the fastest.

Part E) At what time is the culture's weight increasing most rapidly? Explain.

Step 1: Find the second derivative of the logisitc equation

Step 2: Set the second derivative equal to zero and solve for y

Step 3: Set the logistic equation equal to 5 and solve for t

The logisitic growth equation is a fuction of time given by: where A is the limiting factor, k is the constant of proportionality, and C is a constant multiple. To write the logistic equation that models the weight of the culture we must first identify the given values: A = 10 grams, f(0) = 1 gram, f(2) = 2 grams. With these values we can do the following to arrive at the logistic equation that models the weight of the culture.

Part A) Write a logistic equation that models the weight of the bacterial culture

Step 1: Rewrite equation with Given Values

Step 2: Solve for C

Step 3: Plug in C and now Solve for k

Step 4: Plug in C and k to get the final logistic equation

The logistic differential equation is given by: To write the logistic differential equation that models the rate of the culture's weight we must plug in the values of k and A from our logistic equation.Using this equation and Euler's method we can find the approximate weight at time t = 5 hours.

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

Step 1: Plug in k and A into the differential equation and simplify

Step 2: Recall the formula for Euler's method

Step 3: Plug in yold and h and solve for dy/dt to estimate the new weight

Step 4: Repeat this until you can approximate the weight at t = 5 hours.

Step 5: Compare

Exact:

Approx:

To find the culture's weight after 5 hours we simply set t = 5 hours and input it into the logistic equation we got from the Part A. As seen on the right, the culture's weight after 5 hours is 4.571 grams.

Step 1: Recall the logistic equation from Part A

Step 2: Set t = 5 hours and Solve to get the weight

Part B) Find the culture's weight after 5 hours