Question

A large conical tank is positioned so that its vertex is pointed downward. Water drains out from a hole at the vertex. As the water drains, the height of the water (in cm), h (measured from the vertex to the top surface), is always twice the radius of the water's surface (in cm), r. Let v represent the volume of the water in the cone in cm³.

Answer 1

The question regards the mathematical relationship between the height, radius, and volume of a conical tank with draining water. It involves the geometric formula V=1/3*pi*r^2*h and applies to a high school level understanding of volume calculation and fluid dynamics in Mathematics.

Step-by-step explanation:

The student's question pertains to the geometric relationship between the height, radius, and volume of a conical tank from which water is drained. As the water level decreases, the tank maintains the property that the height of the water is always twice the radius of the water's surface. This relationship can be written as h = 2r. Using the formula for the volume of a cone, V = ⅓πr²h, and substituting the given relationship between height and radius, we can define a formula for the volume of water v in the cone as a function of the radius r.

Furthermore, the question refers to several principles of fluid mechanics, such as the pressure change in a cylinder with fluid volume V1 and height hi, as well as concepts like capillary action. These topics illustrate various ways in which fluid volumes and heights are significant in mathematical and physical contexts.