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

[mathdad]

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.

 

[MarkFL]

The population \(P\) at time \(t\) (in minutes) will be:

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

where \(P_0\) is the initial population.

 

[mathdad]

The population \(P\) at time \(t\) (in minutes) will be:

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

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?

 

[HOI]

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).

 

[mathdad]

How do I solve for t?

 

[MarkFL]

How do I solve for t?

Set to population equal to the given amount:

\(\displaystyle 1000\cdot2^{\Large\frac{t}{2}}=500000\)

Divide through by 1000:

\(\displaystyle 2^{\Large\frac{t}{2}}=500\)

Can you finish?

 

[mathdad]

Set to population equal to the given amount:

\(\displaystyle 1000\cdot2^{\Large\frac{t}{2}}=500000\)

Divide through by 1000:

\(\displaystyle 2^{\Large\frac{t}{2}}=500\)

Can you finish?

I defintely can finish.

 

[MarkFL]

I defintely can finish.

I will await your work...

 

[mathdad]

Is this correct?

View attachment 8414

 

[MarkFL]

Taking it from:

\(\displaystyle 2^{\Large\frac{t}{2}}=500\)

Convert from exponential to logarithmic form:

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

\(\displaystyle t=2\log_2(500)\approx17.931568569324174\quad\checkmark\)

 

[mathdad]

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.