How Long to Grow Bacteria Population from 1,000 to 500,000?

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Discussion Overview

The discussion revolves around calculating the time required for a bacteria population to grow from 1,000 to 500,000, given that the population doubles every 2 minutes. Participants explore the mathematical modeling of exponential growth and the steps needed to solve for time.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant asks for help in setting up the equation to determine the time needed for the population growth.
  • Another participant provides the exponential growth formula \(P(t)=P_0\cdot2^{\frac{t}{2}}\), clarifying that \(P_0\) is the initial population.
  • There is a discussion about identifying the exponential nature of the problem based on the doubling time of the population.
  • Participants confirm the initial population as 1,000 and discuss the reasoning behind the exponential equation.
  • Several participants express a desire to solve for \(t\) and share their progress in manipulating the equation.
  • A participant successfully converts the exponential equation to logarithmic form to solve for \(t\), providing an approximate numerical answer.
  • Another participant acknowledges the solution and expresses interest in future problems, indicating a casual engagement with the topic.

Areas of Agreement / Disagreement

Participants generally agree on the approach to solving the problem and the mathematical steps involved, but there is no explicit consensus on the finality of the solution as further discussion could arise.

Contextual Notes

Some participants express uncertainty about the initial setup of the equation and the reasoning behind identifying the problem as exponential growth. The discussion does not resolve all potential assumptions regarding the application of logarithmic functions in this context.

Who May Find This Useful

This discussion may be useful for students or individuals interested in mathematical modeling of population dynamics, particularly in biological contexts, as well as those preparing for standardized tests involving similar problems.

mathdad
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The population of a bacteria culture doubles every 2 minutes. Approximately how many minutes will it take for the population to grow from 1,000 to 500,000 bacteria?

Can someone set up the proper equation needed? I can then work it out.
 
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The population \(P\) at time \(t\) (in minutes) will be:

$$P(t)=P_0\cdot2^{\Large\frac{t}{2}}$$

where \(P_0\) is the initial population.
 
MarkFL said:
The population \(P\) at time \(t\) (in minutes) will be:

$$P(t)=P_0\cdot2^{\Large\frac{t}{2}}$$

where \(P_0\) is the initial population.

1. The initial population is 1000, right?
2. How did you know what to do here? In other words, what words in the application indicated that this is an exponential equation?
 
The key is "The population of a bacteria culture doubles every 2 minutes" so every two minutes the population doubles. Repeated "doubling" is repeated multiplying by 2 and that means a power of 2: 2*2=2^2, 2*2*2= 2^3, etc. In t minutes, there will be t/2 "two minute" intervals so "doubling every two minutes" is 2^{t/2}. And, yes, it is the initial population, 1000 bacteria, that is being "doubled" (multiplied by 2).
 
How do I solve for t?
 
RTCNTC said:
How do I solve for t?

Set to population equal to the given amount:

$$1000\cdot2^{\Large\frac{t}{2}}=500000$$

Divide through by 1000:

$$2^{\Large\frac{t}{2}}=500$$

Can you finish?
 
MarkFL said:
Set to population equal to the given amount:

$$1000\cdot2^{\Large\frac{t}{2}}=500000$$

Divide through by 1000:

$$2^{\Large\frac{t}{2}}=500$$

Can you finish?

I defintely can finish.
 
RTCNTC said:
I defintely can finish.

I will await your work...
 

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  • #10
Taking it from:

$$2^{\Large\frac{t}{2}}=500$$

Convert from exponential to logarithmic form:

$$\frac{t}{2}=\log_2(500)$$

$$t=2\log_2(500)\approx17.931568569324174\quad\checkmark$$
 
  • #11
very good. I will post more GMAT word problems on Wednesday. I am not taking the GMAT. However, the applications are fun to play with.
 

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