# What is the doubling time for bacteria growth in an exponential model?

• KevinL
In summary, the conversation discusses the growth of bacteria in a culture according to an exponential growth model. After the number of bacteria grows from 50 to 1000 in 12 hours, the question is posed to find the number of bacteria present after 18 hours and how long it takes for the number of bacteria to double. The conversation then goes through the calculations, finding that after 18 hours there will be approximately 32017 bacteria and it takes 10.2 hours for the number of bacteria to double. The conversation ends with the discovery of an error in the calculation of the growth rate constant.
KevinL
I feel like I'm doing everything correctly but my answer for (b) doesn't make any sense.

Assume that bacteria in aculture grows according to an exponential growth model. If the number of bacteria grows from 50 to 1000 in 12 hours:

a)How many bacteria will be present after 18 hours?
b)How long does i take for the number of bacteria to double?

A) db/dt = kb, b(0) = 50
b(t)=ce^(kt)

b(0)=50=ce^(k*0)
c=50

b(12)=50e^(k*12) = 1000
e^(k*12)=20
k=12*ln(20)
k=35.94 >>>> .359

Now that I have c and k I can find how much bacteria there is at 18 hours. So:

b(18) = 50e^(.359*18) = 32017

B) I am assuming they mean double as in get to 2000 bacteria. So:
2000=50e^(.359*t)
40=e^(.359t)
ln(40)/.359 = t
10.2 = t

How can it be at 2000 at 10 hours when I already know that 2 hours later its only at 1000? I must have screwed something up.

k=ln(20)/12=0.24964 not k=12*ln(20)

EDIT: yes, that is your only error

Wow that's fairly embarrassing. Thanks for catching that.

## What is an exponential growth model?

An exponential growth model is a mathematical representation of how a quantity increases over time at a rate proportional to its current value. It is characterized by a constant growth rate, meaning that the amount by which the quantity increases each time period is consistent.

## What are the key components of an exponential growth model?

The key components of an exponential growth model are the initial value (or starting amount), the growth rate (or proportionality constant), and the time period over which the growth occurs.

## How is an exponential growth model different from a linear growth model?

In an exponential growth model, the rate of growth (or change) increases over time, whereas in a linear growth model, the rate of growth is constant. This means that the quantity in an exponential growth model increases at a faster and faster rate, while in a linear growth model, it increases at a constant rate.

## What are some real-life examples of exponential growth?

Some real-life examples of exponential growth include population growth, the spread of infectious diseases, and the growth of compound interest in investments. These all exhibit a constant growth rate, resulting in a rapid increase over time.

## What are the limitations of an exponential growth model?

An exponential growth model assumes that the growth rate remains constant, which is often not the case in real-world scenarios. It also does not account for external factors that may affect the growth of a quantity. Additionally, exponential growth is unsustainable in the long term, as resources and space are limited in our world.

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