# Modeling Bacterial Growth with Differential Equations

• QuantumTheory
In summary, The problem in Calculus BC involves a bacteria culture that starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours, there are 8000 bacteria. The task is to find an expression for the number of bacteria after t hours, the number of bacteria after 4 hours, the rate of growth after 4 hours, and when the population will reach 30,000. The problem also involves differential equations and finding the general solution and constants of integration.
QuantumTheory
My friend had to do this problem in Calculus BC. I'm no good at calculus, but I decided it was be fun for someone to figure it out for me.

A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours there are 8000 bacteria.
(a) Find an expression for the number of bacteria after t hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 30,000?

I think it involves differential equations.

QuantumTheory said:
but I decided it was be fun for someone to figure it out for me.

Start with $$y=Ae^{kt}$$

Yeah that amused me too .I'd probably start a little earlier:

Growth proportional to size implies...

$$\frac{dy}{dt}=ky$$

with the intial condition $$y(0)=500$$ and the time t = 3 (in hours!) condition $$y(3)=8000$$.

Your mission, should you choose to accept it, is to find the general solution to the DE above, and the values of k and the constant of integration A.

That's the model, the rest should be ok?

## What is the "Bacteria Growth Problem"?

The "Bacteria Growth Problem" refers to the phenomenon of bacterial populations growing and multiplying in a given environment.

## Why is the "Bacteria Growth Problem" important?

The "Bacteria Growth Problem" is important because it can have significant impacts on human health, agriculture, and the environment. It is also a key area of study in microbiology and other related fields.

## What factors influence bacteria growth?

There are several factors that can influence bacteria growth, including temperature, nutrients, pH levels, oxygen levels, and competition with other microorganisms. These factors can either promote or inhibit bacterial growth.

## How can the "Bacteria Growth Problem" be controlled?

The "Bacteria Growth Problem" can be controlled through various methods such as using antibiotics, disinfectants, and proper sanitation practices. Understanding the factors that influence bacteria growth can also help in controlling it.

## What are the potential consequences of uncontrolled bacteria growth?

Uncontrolled bacteria growth can lead to the spread of infectious diseases, contamination of food and water sources, and damage to ecosystems. It can also result in economic losses in industries such as agriculture and healthcare.

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