$[20$pt$]$ There are a $10-$ mm $-$diameter sphere made of stainless steel $)=15\frac{W}{m}*K,{}_{S.s.}=(4{10}^{-6}\frac{m^2}{s}.$ The sphere is initially at $T_i=20C$ before submerged into a water bath in which water is maintained at $T_{}=50C.$ The convection heat transfer coefficient between the sphere and water is $h=30\frac{W}{m^2}*K.$

a$.$ Determine the temperature at the center of the stainless sphere after $30$ seconds.

b$.$ Determine how long it takes for the sphere to reach $45C.$