Problem
Problem
Water is running into a conical vessel, 15 cm deep and 5 cm in radius, at the rate of . When the water is 6 cm deep, find at what rate is (a) the water level rising? (b) the water surface area increasing? (c) the wetted surface of the vessel increasing?Intuitive explanation
The key is to connect the volume of water flowing in (which you know the rate of) to how fast the depth is changing—and that requires relating the cone's geometry to its water level. Since the cone's radius and depth are proportional, the radius of the water surface grows as the water gets deeper, which means the volume grows faster than you might expect. You'll need to express volume in terms of depth alone, then differentiate with respect to time using the chain rule.
Intuitive explanation
The key is to connect the volume of water flowing in (which you know the rate of) to how fast the depth is changing—and that requires relating the cone's geometry to its water level. Since the cone's radius and depth are proportional, the radius of the water surface grows as the water gets deeper, which means the volume grows faster than you might expect. You'll need to express volume in terms of depth alone, then differentiate with respect to time using the chain rule.