Sand is poured onto the ground to form a right circular conical pile
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Sand is poured onto the ground to form a right circular conical pile. At all times the height of the cone is three times its base radius (that is, h=3r). Sand is added at a rate of 2 m3 per minute. When the height of the pile reaches 6 m, the shape of the pile still satisfies the given relationship.
(a) Express the volume of the conical pile solely as a function of its height h. Show all algebraic steps.
(b) Determine the instantaneous rate of change of the height, dh/dt, at the moment when h=6 m. Include units and justify each step.
(c) Find the corresponding instantaneous rate of change of the radius, dr/dt, at that same moment.
(d) The lateral surface area (excluding the base) of a right circular cone is given by L=πrr2+h2. Compute dL/dt at the moment when h=6 m, and interpret its sign in context.