A solid material that has thermal conductivity \(K\) in kilowatts per meter-kelvin and temperature given at each point by \(w(x, y, z)\) has heat flow given by the vector field \(\mathbf{F}=-K abla w\) and rate of heat flow across a surface \(\mathcal{S}\) within the solid given by \(-K \iint_{\mathcal{S}} abla w d S\).

Find the rate of heat flow out of a sphere of radius \(1 \mathrm{~m}\) inside a large cube of copper \((K=400 \mathrm{~kW} / \mathrm{m}-\mathrm{k})\) with temperature function given by \(w(x, y, z)=20-5\left(x^{2}+y^{2}+z^{2}ight)^{\circ} \mathrm{C}\).