A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 30 – 8(x² + y} + z?) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by –K Vw dS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m - K)). (Use symbolic notation and fractions where needed.) -K Vw dS = kW

A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 30 – 8(x² + y} + z?) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by –K Vw dS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m - K)). (Use symbolic notation and fractions where needed.) -K Vw dS = kW