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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

Answer: b

Explanation: Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).dV, where

dV is the volume of the cone πr3h/3, where r = 1/2π m and h = 4π2 m. On substituting the radius and height in the triple integral, we get 2 units.