I can see a contradiction:
1) varies from 4 to 1 feet
2) y=1.5cos(13.5t)+5.5 --> from 4 to 7
The function f(x) = x cos( ln x ) has indeed infinitely many critical points what is easy to find with the derivation. But it hasn't global min or max because of x multiplying.
"I drew the diagram..." --> That is very well!
Let be r the radius of cylinder. Than try to find out the high (h) of cylinder from the fact: "can be inscribed in a sphere of radius a". Oke, now express the volume of cylinder on depending r:
V(r) = r*r*pi*h(r)
This is the function. Where is its maximum?