Given the above cylinder, set up the minimization problem that will minimize the weight of the cylinder if the torsional shear stress cant exceed $15000$ psi and the maximum deflection can be no larger than $0\mathrm{.}05$ radian.

Shear stress $=T\frac{r}{J}$

Deflection Theta$=$TL$/$GJ

Just form the equation thats the whole question. To minimize weight you reduce the area as there is no information of material use.