Question

wo bowls are filled with water: the paraboloid z = x² + y² and the cone z =√ x² + y². Both are filled to the same height h, and for this particular height h, the volume of water is the same in both bowls. Find h.

Answer 1

The problem requires solving for a common height h at which a paraboloid and a cone, described by their equations, have equal water volumes. Setting up and evaluating the integrals for the volumes and equating them will yield the desired height h.

Step-by-step explanation:

The student's question involves finding the height h at which two geometric shapes—a paraboloid and a cone—both have the same volume when filled with water up to height h. The shapes are given by their respective equations, z = x² + y² for the paraboloid and z = √(x² + y²) for the cone. To solve this problem, we need to set the volumes of the two shapes equal to each other and solve for height h. The volume of a solid of revolution can be found using the method of discs or washers, which integrates the area of a cross-section perpendicular to the axis of revolution. For the paraboloid and cone, this involves setting up and evaluating definite integrals with respect to z in cylindrical coordinates, where z ranges from 0 to h, and the radii of the discs are determined by the shape's equation. The challenge is to find the common height h where their volumes are equal. As the provided references and formulas do not directly address the question, the necessary calculation steps involve standard integration techniques applied to the problem's context and finding the particular height where the two volumes match. It is not a direct application of formulas such as Torricelli's theorem or the kinematic equation for falling objects.