A water tank is an inverted right circular cone with a height of 12 m and a top…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A water tank is an inverted right circular cone with a height of 12 m and a top radius of 4 m.
Water is being pumped into the tank at a constant rate of 0.60 m3 min−1 while simultaneously draining through a valve at 0.15 m3 min−1.
The depth of the water at time t minutes is h(t), measured vertically from the tip of the cone (its lowest point).
Assume the water surface is always a circular cross-section whose radius varies with h.
(a) Express the radius of the water surface, r, as a function of the current depth h. Explain your reasoning.
(b) Obtain a formula for the instantaneous rate of change of the water depth, dtdh, in terms of h. Clearly show how you relate volume change to depth change.
(c) Calculate dtdh when the water depth is 5.0 m. Include units and interpret the result.
(d) Find the rate at which the radius of the water surface is changing at the same instant (h=5.0 m). Indicate whether the surface is expanding or contracting and justify your answer.