Final answer:
To find the intersection of the helix with the paraboloid, we equate the z-component of the helix to the equation of the paraboloid, resulting in a transcendental equation with potentially multiple solutions.
Step-by-step explanation:
The original question seems to be a mix of unrelated mathematical expressions, which do not provide a clear context for a problem. However, focusing on the task at hand, we need to find the intersection of the helix r(t) = ⟨cos(πt), sin(πt), t⟩ and the paraboloid z = x²y². To find the intersection, we equate the z-component of the helix to the equation of the paraboloid.
Substituting the x and y components from the vector equation of the helix into the paraboloid equation, we get: t = cos(πt)² sin(πt)². Solving this equation will give us the t-values at which the intersection occurs. However, without further specifications, such as interval constraints, this transcendental equation can have multiple solutions, which generally require numerical methods to solve.
Thus, the helix and paraboloid may intersect at several points, and additional information or context is required to determine the specific points of intersection.