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# calculus populations?

A bacteria culture starts with 980 bacteria and grows at a rate proportional to its size. After 4 hours there will be 3920 bacteria. Any work shown would be a great help, thanks!

(a) Express the population after t hours as a function of t.

b) What will be the population after 3 hours?

c.) How long will it take for the population to reach 2450 ?

### 2 Answers

- 1 decade agoFavorite Answer
This is a borderline Differential Equation problem, so I may write it more complicated then you need to know.

A)

Dp/dt=kp (In english, the change of population is proportional to the population).

Solving this differential equation gives you:

P(t)=P_0*e^(kt)

This means population at time t is equal to the initial population times some growth factor.

b) We know P_0 is 980 from the problem. And we know at a certain time (when t=4) that the population is 3920. So we just plug them into the above formula and solve for K.

c) We should now have a K value from doing the above algebra. But now you need to find the time (t) when the population=2450. So then substitute in your k value into the equation from part A so it looks like

2450=980*e^(kt)

Figuring out T is mostly just algebra at this point.

- 6 years ago
A bacteria population starts with 980 bacteria and grows at at rate proportional to its size. After 4 hours there will be 3920 bacteria. Express the population after t hours as a function of t.