Doubling time!

• Jan 23rd 2008, 02:42 PM
mathlete
Doubling time!
If the size of a bacteria colony doubles in 6 hours, how long will it take for the number of bacteria to triple? Give your answer in exact form and decimal form.

Exact:

Decimal (nearest hundreth):

ok so i know the equation that i have to use, i believe its,

2=(1+r/100)^t

and then for the tripleing time you just find, r, and plug it in...can someone let me know if im on the right track..thanks

mathlete
• Jan 23rd 2008, 08:50 PM
earboth
Quote:

Originally Posted by mathlete
If the size of a bacteria colony doubles in 6 hours, how long will it take for the number of bacteria to triple? Give your answer in exact form and decimal form.
...

Hi,

1. I assume exponential growth
2. let a be the initial amount of bacteria

With exponential growth you use the equation:

$\displaystyle A(t) = A(0) \cdot e^{k \cdot t}$ with A(t) = amount at the time t, and A(0) = initial value.

From your problem you know: A(6) = 2a

$\displaystyle 2a = a \cdot e^{k \cdot 6}~\iff~2 = e^{k \cdot 6}$$\displaystyle ~\iff~ \ln(2)= k\cdot 6~\iff~ k= \frac16 \cdot \ln(2)$

Use these results to calculate A(t) = 3a

$\displaystyle 3a = a \cdot e^{\frac16 \cdot \ln(2) \cdot t}~\iff~3 = e^{\frac16 \cdot \ln(2) \cdot t}$ $\displaystyle ~\iff~ \ln(3)= \frac16 \cdot \ln(2) \cdot t ~\iff~ t= 6 \cdot \frac{\ln(3)}{\ln(2)}$

Use a calculator to get the approximative value of $\displaystyle t \approx 9.50977... \approx 9.51$