A bacteria triples its original population in 40 hours (A=3A0). How big will its population be in 120 hours? his problem requires two main steps. First, we must find the unknown rate, k. Then, we use that value of k to help us find the unknown number of bacteria.
Identify the variables in the formula.
AA0ktA=3A0=?=?=40 hours=A0ekt
Substitute the values in the formula.
3A0=A0ek⋅40
Solve for k. Divide each side by A0.
3A0A0=e40k
Take the natural log of each side.
ln3=lne40k
Use the power property.
ln3=40klne
Simplify.
ln3=40k
Divide each side by 40.
ln340=k
Approximate the answer.
k≈0.027
We use this rate of growth to predict the number of bacteria there will be in 120 hours.
AA0ktA=3A0=?=ln340=120 hours=A0ekt
Substitute in the values.
A=A0eln340⋅120
Evaluate.
A=27A0

At this rate of growth, we can expect the population to be 27 times as large as the original population.