Q.A right circular cone has height 9 cms and base radius 5 cms. It is inverted and water is poured into it. If at any instant the water level rises at the rate of $\dfrac{\pi}{A}$ cms/sec, where $A$ is the area of the water surface at that instant, show that the vessel will be full in 75 seconds.

Beyond growth, decay and cooling, a differential equation can capture any situation where a rate of change is described in words and the underlying quantity is the actual object of interest — for example, water rising in a cone at a rate tied to the wetted curved-surface area, a raindrop evaporating at a rate proportional to its own surface area, a balloon's volume inflating at a constant rate, or a person's assets shrinking at a rate proportional to the square root of the current assets. Solving these problems has two distinct stages: first, translating the verbal description into a differential equation by identifying which rate (dV/dt, dh/dt, dr/dt, dA/dt, ...) is being described and what it is stated to be proportional (or equal) to, often requiring an extra geometric relation (such as similar triangles linking a cone's water radius to its height, or t …