As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around 30 watts per cubic kilometer. (a watt is a rate of heat production.) the heat then flows to the earth's surface where it is lost to space. let f^ (x, y,z) denote the rate of flow of heat measured in watts per square kilometer. by definition, the flux of f^ across a surface is the quantity of heat flowing through the surface per unit of time.

a. suppose that the actual heat generation is 28 w/km3. what is the value of div f^
b. assume the heat flows outward symmetrically. verify that f^= αr^ where r=xi^+yj^+zk^ and α is a suitable constant, satisfies the given conditions. find α.
c. let t(x, y,z) denote the temperature inside the earth. heat flows according to the equation. f^=−k grad t where k is a constant. if t is in °c, then k = 28,000°c/km. assuming the earth is a sphere with radius 6400 km and surface temperature 20°c, what is the temperature at the center?