It sounds like I need derivatives and rates of change but this course does not involve calculus...so how would I go about this mathematically?

set up a system of two equations.

let \(\displaystyle m\) = block mass

\(\displaystyle I\) = cylinder's rotational inertia

\(\displaystyle T\) = tension in the string

\(\displaystyle R\) = cylinder radius

\(\displaystyle a\) = linear acceleration of the block

\(\displaystyle \alpha\) = angular acceleration of the cylinder

net force acting on the block ...

\(\displaystyle mg - T = ma\)

net torque acting on the cylinder ...

\(\displaystyle TR = I\alpha\)

the "tie" between the two equations is \(\displaystyle \alpha = \frac{a}{R}\)

solve the system for the linear acceleration of the block and the rest should be rather easy to do.