$7.$ A copper cube $10\mathrm{.}0$ cm on a side is heated to $100$ C$.$ The block is placed

on a surface that is kept at $0$ C$.$ The sides of the block are insulated, so

the normal derivatives on the sides are zero. Heat flows from the top of

the block to the air governed by the gradient uz $=$

$10$C$/$m$.$ Determine

the temperature of the block at its center after $1\mathrm{.}0$ minutes. Consider the

following hints:

a$.$ This is a heat conduction problem with nonhomogeneous boundary

conditions. Assume u$($x$,$ y$,$ z$,$ t$)=$ v$($x$,$ y$,$ z$,$ t$)+$ f$($z$),$ where v$($x$,$ y$,$ z$,$ t$)$

satisfies homogeneous boundary conditions. Find v$($x$,$ y$,$ z$,$ t$)$ and f$($z$).$

b$.$ In order to get a numerical value for the temperature, you will need

the thermal diffusivity, which is given by k $=$ K

c

p $,$ where K is the ther

mal conductivity, is the density, and cp is the specific heat capacity.

Look up any needed properties of copper.