Final Answer:
The capacity of the vessel can be expressed in terms of π (pi) and is approximately 297π cubic inches.
Step-by-step explanation:
To find the capacity of the vessel, which is in the form of a cone mounted by a hollow cylinder, we need to calculate the volumes of both the cone and the cylinder separately and then subtract the volume of the cone from the volume of the cylinder.
Let's denote the radius of the cone and the cylinder as 'r' (half of the diameter). The diameter is given as 6 inches, so the radius 'r' is 3 inches. The height of the vessel (which is the height of the cone and the cylinder combined) is given as 11 inches.
Volume of the Cone (Vcone): The formula for the volume of a cone is Vcone = (1/3)πr²h, where 'r' is the radius and 'h' is the height. Substituting the values, we get Vcone = (1/3)π(3)²(11).
Volume of the Cylinder (Vcylinder): The formula for the volume of a cylinder is Vcylinder = πr²h, where 'r' is the radius and 'h' is the height. Substituting the values, we get Vcylinder = π(3)²(11).
Finally, we subtract the volume of the cone from the volume of the cylinder: V = Vcylinder - Vcone = π(3)²(11) - (1/3)π(3)²(11), which simplifies to approximately 297π cubic inches.