$5.$ A cylindrical copper rod, initially at a uniform temperature of $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ is suddenly immersed in a large quiescent water bath maintained at $\backslash (25^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The rod measures $10$ cm in diameter and $2$ m in length. The top and bottom surfaces are carefully thermally insulated, while the curved surface is in direct contact with the cooling water. Copper properties remain approximately constant as their values at $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ The effective convection heat transfer coefficient at the immersed surface is $\backslash (50\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot$\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$

After $30$ minutes, determine:

a$.$ The temperature at the center of the rod.

b$.$ The temperature halfway between the center and the outer surface, measured along the radius.