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Karolina [17]
3 years ago
6

A biologist observed that a certain bacterial colony obeys the population growth law and that the colony triples every 4 hours.

Mathematics
1 answer:
MrRa [10]3 years ago
4 0

Answer:

a) P(t) = 2e^{0.275t}

b) 54.225 square centimeters.

c) 2.52 hours

Step-by-step explanation:

The population growth law is:

P(t) = P_{0}e^{rt}

In which P(t) is the population after t hours, P_{0} is the initial population and r is the growth rate, as a decimal.

In this problem, we have that:

The colony occupied 2 square centimeters initially, so P_{0} = 2

The colony triples every 4 hours. So

P(4) = 3P_{0} = 6

(a) An expression for the size P(t) of the colony at any time t.

We have to find the value of r. We can do this by using the P(4) equation.

P(t) = P_{0}e^{rt}

6 = 2e^{4r}

e^{4r} = 3

Applying ln to both sides, we get:

4r = 1.1

r = 0.275

So

P(t) = 2e^{0.275t}

(b) The area occupied by the colony after 12 hours.

P(t) = 2e^{0.275t}

P(12) = 2e^{0.275*12}

P(12) = 54.225

(c) The doubling time for the colony?

t when P(t) = 2P_{0} = 2*2 = 4.

P(t) = 2e^{0.275t}

4 = 2e^{0.275t}

e^{0.275t} = 2

Applying ln to both sides

0.275t = 0.6931

t = 2.52

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