Question

A bacteria culture starts with 880 bacteria and grows at a rate

proportional to its size. After 5 hours there will be 4400 bacteria.
(a) Express the population P after t hours as a function of t.
P(t)=
(b) What will be the population after 7 hours?
bacteria
(c) How long will it take for the population to reach 2570? Give
your answer accurate to at least 2 decimal places.
hours

Answer 1

(a) We can use the formula for exponential growth, which is P(t) = P0e^(kt), where P0 is the initial population, k is the growth rate constant, and e is the base of the natural logarithm. Since the population is proportional to its size, we can write:

P(t) = P0e^(kt)

We know that P(0) = 880 and P(5) = 4400, so we can use these values to find k:

4400 = 880e^(5k)

e^(5k) = 5

5k = ln(5)

k = ln(5)/5

Substituting this value of k into the formula, we get:

P(t) = 880e^(t*ln(5)/5)

(b) To find the population after 7 hours, we can simply plug in t = 7 into the formula we found in part (a):

P(7) = 880e^(7*ln(5)/5) ≈ 11036.29

So there will be approximately 11036 bacteria after 7 hours.

(c) To find how long it takes for the population to reach 2570, we can set P(t) = 2570 and solve for t:

2570 = 880e^(tln(5)/5)

e^(tln(5)/5) = 2570/880

tln(5)/5 = ln(2570/880)

t = 5ln(2570/880)/ln(5)

Using a calculator, we get:

t ≈ 2.51 hours (rounded to 2 decimal places)

So it will take approximately 2.51 hours for the population to reach 2570 bacteria.

Explanation: