Solving Polynomial Equations and Inequalities The distance, in km, of a ship from its harbour is modeled by the function 
  $d(t)= -3t^3 + 3t^2 + 18t$
  where $t$ is the time elapsed in hours since departure from the harbour.


a) When does the ship return to harbour? 

b) Estimate the time that the ship begins its return trip back to the harbour. 

c) When is the ship $24$ km from the harbour? 

I believe I have figured the first part of this question out.  I substituted values for "$t$" until the equation ended with a value of zero.  This value was $d(3)$.  Meaning that it takes the ship 3 hours to reach the harbour.  The other questions I am just not sure how to approach.  c) I began by substituting $24$ as my distance, but I am not sure what to do after that.
As of your answer to a), I wouldn't recommend that, instead do:
$$d(t)=0\implies -3t^3+3t^2+18t=0\implies 3t(-t^2+t+6)=0\implies t=0\text{ or }-t^2+t+6=0\implies t=0\text{ or }t=3\text{ or }t=-2$$
Do you understand how we found these solutions? It should be clear that from the above we can conclude that the ship returns at $t=3$

As of b), when would you expect this?

As of c), we have to solve $d(t)=24$. Now, our method of a) doesn't work anymore, since we have to solve $$-3t^3+3t^2+18t-24=0$$
we can't factor out a term $t$. Substituting some standard values is the only solution (actually there are ways to solve this, but I don't wan't to bother you with that!). And then we find $t=1$ is a solution.
As of your answer to a), I wouldn't recommend that, instead do:
$$d(t)=0\implies -3t^3+3t^2+18t=0\implies 3t(-t^2+t+6)=0\implies t=0\text{ or }-t^2+t+6=0\implies t=0\text{ or }t=3\text{ or }t=-2$$
Do you understand how we found these solutions? It should be clear that from the above we can conclude that the ship returns at $t=3$

As of b), when would you expect this?

As of c), we have to solve $d(t)=24$. Now, our method of a) doesn't work anymore, since we have to solve $$-3t^3+3t^2+18t-24=0$$
we can't factor out a term $t$. Substituting some standard values is the only solution (actually there are ways to solve this, but I don't wan't to bother you with that!). And then we find $t=1$ is a solution.