Question

The number of algae in a tub in a labratory increases by 10% each hour. The initial population, i.e. the population at t = 0, is 500 algae.(a) Determine a function f(t), which describes the number of algae at a given time t, t in hours.(b) What is the population at t = 2 hours?(c) What is the population at t = 4 hours?

Answer 1

a) Let's say initial population is po and p = p(t) is the function that describes that population at time t. If it increases 10% each hour then we can write:

t = 0

p = po

t = 1

p = po + 0.1 . po

p = (1.1)¹ . po

t = 2

p = 1.1 . (1.1 . po)

p = (1.1)² . po

t = 3

p = (1.1)³ . po

and so on

So it has an exponential growth and we can write the function as follows:

p(t) = po . (1.1)^t

p(t) = 500 . (1.1)^t

Answer: p(t) = 500 . (1.1)^t

b)

We want the population for t = 2 hours, then:

p(t) = 500 . (1.1)^t

p(2) = 500 . (1.1)^2

p(2) = 500 . (1.21)

p(2) = 605

Answer: the population at t = 2 hours is 605 algae.

c)

Let's plug t = 4 in our function again:

p(t) = 500 . (1.1)^t

p(4) = 500 . (1.1)^4

p(4) = 500 . (1.1)² . (1.1)²

p(4) = 500 . (1.21) . (1.21)

p(4) = 500 . (1.21)²

p(4) = 732.05

Answer: the population at t = 4 hours is 732 algae.