MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A conical reservoir is upright (vertex down). The reservoir has a height of 15 m and a top radius of 9 m. Water is pumped into the reservoir at a constant rate of 5 m3/min. Simultaneously, evaporation causes the water depth to decrease at a constant rate of 0.01 m/min.
(a) Obtain a formula that relates the radius r of the water’s surface to its depth h.
(b) Express the volume of water V in the reservoir as a function of h only.
(c) Write an equation that relates dtdV to dtdh at any moment.
(d) When the water depth is 6 m, compute the instantaneous rate of change of the volume of water in the reservoir due solely to the changing depth (ignoring the pumping).
(e) Using your results, determine the overall rate at which the water depth is changing when h=6 m, taking both pumping and evaporation into account, and indicate whether the depth is increasing or decreasing at that instant.