Re: Microbiology Hwk Problem, Calculating generation time

[shawonna23]

Calculate the time it will take to increase the cell number from 10^4 CFU/ml to 10^8 CFU/ml assuming a generation time of 1.5 hr.


2=e^ut where t is the generation time...solve for u.

x=xo*e^ut where t is time

For the 1st equation:
ln2=lne(ut)
ln2/t=u
ln2/1.5=u
u=0.46 l/hr

For the 2nd equation:
x-xo=e^ut
ln(x-xo)=lne(ut)
ln(10^8-10^4)/0.46=t

I don't know how to solve the 2nd equation.

 

[Moonbear]

Your notation is a bit different from what I'm accustomed to, and typed some parts strangely, so I'm not quite sure if you understand what you're doing or not, because you seem to have gone ahead and corrected the mistakes in the next step.

For example, you wrote:
ln2=lne(ut)
but that should be ln2=ln(e^ut)
Yet, to get to this step:
ln2/t=u
you must have known that (unless you're just copying steps from a notebook and aren't following the math here, in which case, you should review how to work with natural logs).

I'm not sure why you have units shown for your solution for u in the first part of your solution. There should be no units...it's a rate constant.

By the way, the first equation actually is identical to the second equation, it's just already been simplified for the first doubling. You're using the known doubling in one generation to calculate the rate constant there, then using the rate constant to solve for the final time in your second equation.

However, the main mistake you've made is after you insert your solution for the rate constant into the second equation. Somehow, you've subtracted the original population size rather than divided by it from both sides of the equation. You have the same error in notation for ln(e^ut) as well, so be really careful you're doing that right.

If you divide rather than subtract, I think you'll find you will quickly get to the right solution. In other words, your error is algebraic, not in the biology. See if you can get that on your own now. Make sure you understand the algebra at each step so you're not just doing steps from memory, but actually know how to manipulate the equations to find each variable.

 

[Blueshift5]

Hello,

To solve the second equation, you can use the properties of logarithms to rewrite the equation as:

ln(10^8-10^4) = ln(e^ut)

Then, you can use the inverse of the natural logarithm (e^x) to simplify further:

10^8-10^4 = e^ut

Finally, you can solve for t by dividing both sides by u and taking the natural logarithm of both sides:

ln(10^8-10^4)/u = t

Substituting the value of u (0.46 l/hr) that we found in the first equation, we get:

ln(10^8-10^4)/0.46 = t

This will give you the time it will take to increase the cell number from 10^4 CFU/ml to 10^8 CFU/ml. I hope this helps! Let me know if you have any further questions.