For a function given by F = 4x i + 7y j +z k, the divergence theorem evaluates to which of the values given, if the surface considered is a cone of radius 1/2π m and height 4π2 m.

To apply the divergence theorem to the vector field ( mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k} ), we first need to calculate the divergence of ( mathbf{F} ).

The divergence ( nabla cdot mathbf{F} ) is calculated as follows:

[

nabla cdot mathbf{F} = frac{partial}{partial x}(4x) + frac{partial}{partial y}(7y) + frac{partial}{partial z}(z)

]

[

nabla cdot mathbf{F} = 4 + 7 + 1 = 12

]

Next, we need to find the volume of the cone with radius ( r = frac{1}{2pi} ) m and height ( h = 4pi^2 ) m. The volume ( V ) of a cone is given by the formula:

[

V = frac{1}{3} pi r^2 h

]

Substituting the values we have:

[

V = frac{1}{3} pi left(frac{1}{2pi}right)^2 cdot 4pi^2

]

[

= frac{1}{3} pi cdot frac{1