To determine whether the fluid is incompressible or compressible and whether the flow is rotational or irrotational, we need to analyze the given velocity field \(\vec{V}(x, y) = 3 \hat{i} + (5x) \hat{j}\).

Check for Incompressibility: A fluid flow is incompressible if the divergence of the velocity field is zero. The divergence in two dimensions is given by:

$\nabla \cdot \vec{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y}$

For the given velocity field, where \(V_x = 3\) and \(V_y = 5x\), the divergence becomes:

$\nabla \cdot \vec{V} = \frac{\partial 3}{\partial x} + \frac{\partial (5x)}{\partial y} = 0 + 0 = 0$

Since the divergence is zero, the fluid is incompressible.

Check for Rotationality: A flow is rotational if the curl of the velocity field is non-zero. In two dimensions, we only need to consider the k-component of the curl:

$\nabla \times \vec{V} = \left(\frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\right) \hat{k}$

For the given velocity field:

$\nabla \times \vec{V} = \frac{\partial (5x)}{\partial x} - \frac{\partial 3}{\partial y} = 5 - 0 = 5$

Since the curl is non-zero, the flow is rotational.

Therefore, the correct statement is "The fluid is incompressible and the flow is rotational."