Question

A bacterial culture grows exponentially according to the function n(t)=n(e^n), where n(t) is the quantity after t hours and n is the original quantity. If the culture grows from 2 grams to 128 grams in 3 hours, how many grams where there after 2 hours?

Answer 1

`n(t=2) = 2 e^(1.386294361(2))=32 grams`

So then after 2 hours we will have 32 grams.

Explanation:

For this case we have the followin exponential model:

`n(t) = n e^(rt)`

n(t) is the quantity after t hours, n is the original quantityand t represent the hours and r the rate constant.

For this case we know that n(0) = 2 grams and n(3) = 128 grams and we want to find n(2)=?

From the initial condition we know that n = 2, and we have the model like this:

`n(t) = 2 e^(rt)`

Now if we apply the other conditionn(3) = 128 we got:

`128 = 2 e^(3r)`

If we divide both sides by 2 we got:

`64= e^(3r)`

If we apply natural log for both sides we got:

`ln(64) = 3r`

`r = (ln(64))/(3)=1.386294361`

And our model is this one:

`n(t) = 2 e^(1.386294361t)`

And if we replace t = 2 hours we got:

`n(t=2) = 2 e^(1.386294361(2))=32 grams`

So then after 2 hours we will have 32 grams.