Rumour problem (logistic differential equations) - somewhat urgent

Hello. I'm currently stuck on a problem that I can't seem to work out. It's quite complex and difficult (for me anyway) and it would be great if you could help me out. Below is the background required:

Consider a population where members know the rumour and meet one of members who doesn't know the rumour.

The rate of change in the population who know the rumour is hence, given by:

where is some constant.

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Here's the question I'm stuck with:

A town has N residents. At 8 am, a rumour begins with 2 people (spread by a logistic growth). At 9 am, 22 know the rumour. At 12 pm, half the population know the rumour. Find the population and k.

First, I transformed it into a differential equation and proceeded to try and solve it:

=

=

=

=

=

Now I'm stuck...It sounds as easy as plugging in the values where:

And solve it simultaneously. However, I can't seem to get it (did I do something wrong?)

If you could assist me from this point on, it would definitely be appreciated.

The answers given by the textbook is:

k = 0.00008169

Popluation = 29360

Thanks.