My question is about b) which is asking for the maximum height. Attached as well is the solutions manual:

It did take me a while to understand a) but I see that as h goes to zero that b-32t is left so for now I have to accept it and move on.

As for b) the solutions manual says that v(t) = 0 and that velocity is t =b/32.

OK fine it must be the answer however at this point b is not known or is it? I tried graphing the equation b-32t however I am finding my graphing calculator only allows for one variable or maybe I just havn't figured out how to work another variable b in.

The solutions manual goes on to describe s(b/32) = b^2/64 which also I am working on understanding but I just don't see how that was arrived at either.

Perhaps I could get some explanation?

Thanks in advance...

P.S. I think both terms were multiplied by 1/32 and then solved for t to get b/32?