### Video Transcript

Suppose a populationβs growth is
governed by the logistic equation dπ by dπ‘ equals 0.07π multiplied by one minus
π over 900, where π of zero is equal to 50. Write the formula for π of π‘.

Writing π of π‘ means that we need
to find the solution to this logistic equation. We can begin by writing down its
general form. We know that for the logistic
equation dπ by dπ‘ equals ππ multiplied by one minus π over πΏ that its solution
is given by π equals πΏ over one plus π΄π to the negative ππ‘, where π΄ is equal
to πΏ minus π nought over π nought. Here π represents the growth rate
of the population. πΏ is the carrying capacity. And π nought is the initial
population. We can identify each of these
values from the information given in the question.

First, we see that π is equal to
0.07 and πΏ is equal to 900. Weβre also told that π zero is
equal to 50. So we can fill in each of the
values in the general solution. Letβs work out π΄ first of all. π΄ is equal to πΏ minus π nought
over π nought. Thatβs 900 minus 50 over 50 or 850
over 50, which is equal to 17.

Now we can substitute into the
general form of the solution. π is equal to πΏ β thatβs 900 β
over one plus π΄ β thatβs 17 β π to the power of negative π β thatβs negative 0.7
β π‘. So we have our solution for π or
π of π‘. Itβs equal to 900 over one plus
17π to the power of negative 0.07π‘.