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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.
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