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 find the divergence of the vector field (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}) and apply the divergence theorem to the specific geometry given (a cone with radius (frac{1}{2pi}) m and height (4pi^2) m), we first calculate the divergence of (mathbf{F}).

The divergence of a vector field (mathbf{F} = Pmathbf{i} + Qmathbf{j} + Rmathbf{k}) is given by:

[ nabla cdot mathbf{F} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]

For (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}), we have:

[ nabla cdot mathbf{F} = frac{partial (4x)}{partial x} + frac{partial (7y)}{partial y} + frac{partial (z)}{partial z} = 4 + 7 + 1 = 12 ]

Now, the divergence theorem states that for a vector field (mathbf{F}