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Alexandra [31]
3 years ago
13

A bacteria culture starts with 660 bacteria and grows at a rate proportional to its size. After 6 hours there will be 3960 bacte

ria. (a) Express the population after t hours as a function of t. population: (function of t) (b) What will be the population after 3 hours?
Mathematics
1 answer:
scoray [572]3 years ago
4 0

Answer:

a) P(t) = 660 e^{\frac{ln(6)}{6} t}

b) P(t=3) = 660 e^{\frac{ln(6)}{6}*3} = 1616.663 \approx 1617

Step-by-step explanation:

For this case we have the following info:

P(0) = 660 represent the initial amount of bacteria

P(6) = 3960 represent the population of bacteria after 6 hours

Part a

For this case we use the proportional model given by:

\frac{dP}{dt}= kP

Where P represent the population at time t and k is a constant of proportionality.

We can rewrite the model like this:

\frac{dP}{P} = k dt

And if we integrate both sides we got:

ln |P|= kt + C

Now if we apply exponential we got:

P = e^{kt +c}= e^{kt} e^C

P(t)= P_o e^{kt}

For this case using the initial condition:

660 = P_o e^{0k}, P_o = 660

We have the following model:

P(t) = 660 e^{kt}

Using the second condition P(6) = 3960 we have this:

3960 = 660e^{6k}

We can divide both sides by 660 and we got:

6= e^{6k}

Now we can apply natural log on both sides and we got:

ln (6)= 6k

k = \frac{ln(6)}{6}=0.2986265782

So then the model would be given by:

P(t) = 660 e^{\frac{ln(6)}{6} t}

Part b

For this case we just need to replace t=3 into the model and we got:

P(t=3) = 660 e^{\frac{ln(6)}{6}*3} = 1616.663 \approx 1617

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