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 (measured from the vertex to the top surface) is always twice the radius of the waters surface.

Draw and label a diagram of the situation, defining any variables you use

Answer 1

h=2r, where h is the height of the water and r is the radius of the water's surface.

In this conical tank scenario, let's denote the height of the water from the vertex to the top surface as h and the radius of the water's surface as r. According to the given condition, h is consistently twice the value of r, which can be expressed as h=2r.

This relationship between the height and radius ensures that the geometry of the water column remains in a specific proportion as it drains out from the hole at the vertex of the conical tank. The conical shape implies that as water decreases, maintaining a 2:1 ratio between the height and radius, the tank maintains a symmetrical and predictable form. This mathematical relationship helps describe the dynamic geometry of the water column during the draining process in a simple and consistent manner.