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)

b)

c)

Step-by-step explanation:

(a) Suppose that the actual heat generation is 27W/km3 What is the value of div F? div F- Include units)

For this case the value for div F correspond to the generation of heat.

(b) Assume the heat flows outward symmetrically. Verify that where . Find a α, (Include units.)

For this case we can satisfy this condition:

And since we have the value for the we can find the value of like this:

(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=27000 C/km. Assuming the earth is a sphere with radius 6400 km and surface temperature 20°C, what is the temperature at the center? 27,0 C/km.

For this case we have this:

And grad T represent the direction of the greatest decrease related to the temperature.

So we have this equation:

And we can solve for grad T like this:

Andif we integrate in order so remove the gradient on both sides we got:

For our case we have the following condition:

And we can solve for C like this:

So then our equation would be given by:

And for our case at the center we have that

And we got: