Final answer:
The volume of water in the inverted cone, the depth of water from the vertex of the second cone, and the volume of the frustum-bucket can be calculated using geometrical formulas for cones and frustums, including the ratio of dimensions and subtracting heights for specific cases.
Step-by-step explanation:
To calculate the volume of water in the given inverted cone of height 15 cm and base radius 4 cm containing water to a depth of 10 cm, we can use the formula for the volume of a cone, V = \( \frac{1}{3} \pi r^2 h \), where r is the radius and h is the height of the water level. However, since the cone is not completely filled, we need to find the ratio of the dimensions of the water level to the cone.
To calculate the depth of water in a cone, measured from the vertex, for an inverted cone of height 12 cm and base radius 6 cm containing 20 cm of water, we need to subtract the height of the water column from the total height of the cone (12 cm) to get the depth from the vertex.
For the volume of the bucket shaped as a frustum with diameters of 10 cm and 4 cm at its ends and a depth of 3 cm, we apply the formula V = \( \frac{1}{3} \pi h (r_1^2 + r_1 r_2 + r_2^2) \), where r_1 and r_2 are the radii of the two circular ends of the frustum and h is the height of the frustum.