MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A conical water tower, vertex down, has a height of 20 m and a base radius of 8 m. Water flows into the tower from an external source at a constant rate of 3.0 m3\,/min. Simultaneously, a small leak at the vertex allows water to escape at a rate (in m3\,/min) given by 0.25h, where h is the depth of the water in meters at that instant.
Assume at time t=0 minutes the depth of the water is 6 m.
(a) Derive an expression for the instantaneous rate of change of the water depth dtdh as a function of h. Include units in your expression.
(b) Determine the exact depth h (if any) at which the water level would remain momentarily constant. Justify your answer.
(c) For the initial depth of 6 m, decide whether the water level is rising or falling and compute dtdh at that moment.
(d) Explain, using calculus-based reasoning, whether the water level will ever stabilize permanently, or if it will ultimately rise to fill the tank or drain completely. Support your conclusion with appropriate mathematical arguments.