Question

The population of a colony of birds at atime, t months where 0 13 years ago

Answer 1

This P(t) is the function given as used in the model. We see that when t=0 and t=3, P(t)=300 birds. But we intereted in how the function behaves in between those times.
To do that we need to find the maximum on on the graph of P(t)
P(t) = 20t(9-t^2)+300
=180t -20t^3 +300
p'(t) =180 -60t^2
to find min or max is when P'(t) = 0
0 = 180-60t^2
3= t^2
t= +/- 3^0.5 we can't have t= -3^0.5 so it has to be t =3^0.5
We now put t=3^0.5 back into the model function, should give 507 birds,

Answer 2

In this case we have been given the function P(t) and the time constraint. We see that at t=0 and t=3 is 300 birds so the graph cuts y at 300 and x at 300. we can assume the graph is up between the times but we don't really know if you weren't given the max = 507 birds. Even if we didn't know that, we would have worked it out by using general calculus rules. Hope this helps.