# Thread: Help with calculus problem

1. ## Help with calculus problem

In a certain culture of bacteria, the number of bacteria increased sixfold in 10h. Assuming natural growth, how long did it take for their number to double?

2. Originally Posted by Abalo
In a certain culture of bacteria, the number of bacteria increased sixfold in 10h. Assuming natural growth, how long did it take for their number to double?
This is classic exponential growth so use the equation A(t) = A_0e^{kt}

where:

• $A(t)$ = Amount at time t
• $A_0$ = Amount at time 0/initial amount
• $k$ = growth constant
• $t$ = time

By definition the amount at time 0 is equal to $A_0$

If the increase is sixfold after ten hours we get the following:

$6A_0 = A_0e^{10k}$ which is equal to $6 = e^{10k}$

From there you can find k

Once you know your value of k find t using the rearranged equation below and that $A_2 =2A_0= A_0e^{kt}$

$t = \frac{1}{k} \left[ \ln (A_t) - \ln (A_0)\right]$

### in a certain culture of bacteria the number of bacteria increased sixfold in 10h

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