A storage rack is to be designed to hold the paper roll as shown in $($Figure $1).$ The paper roll has a $1\mathrm{.}50$m outside diameter $($OD$),$ a $0\mathrm{.}22$m inner diameter $($ID$),$ a $3\mathrm{.}48$mlength, and has a density of $984$ kg$/$m$^3.$ Take the static safety factor of $1\mathrm{.}5.$

The mandrel is solid and inserts halfway into the paper roll.

Assume that the paper roll's weight creates a concentrated load acting at the tip of the mandrel, while the upper reaction the stanchion exerts on the mandrel is uniformly distributed all over the length of the hole, and the bottom reaction is concentrated at the right end of the hole. Consider bending and transverse shear strength.

Part A

If the beam is a ductile material with S$\_$y $=300$ MPa, determine suitable value for dimension $"$a$"$ in the $($Figure $1).$

Express your answer to three significant figures and include the appropriate units.

Part B

If the beam is a ductile material with S$\_$y $=300$MPa, determine suitable value for dimension $"$b$/\text{'}$ in the $($Figure $1).$

Express your answer to three significant figures and include the appropriate units.