Sand is poured at a constant rate of into a conical pile whose base radius is always…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Sand is poured at a constant rate of

0.08\,\text{m}^{3}\!/\text{min}

into a conical pile whose base radius is always exactly

\tfrac{5}{12}

of its height. At the moment when the pile is

1.5\,\text{m}

tall, the height is increasing. (1) Part 1: (a) Express the radius of the pile in terms of its height. (2) Part 2: (b) Write the volume of the pile solely as a function of its height. (3) Part 3: (c) Calculate the instantaneous rate at which the height of the pile is increasing, in meters per minute, when the height is

1.5\,\text{m}

. Show all algebraic steps and include appropriate units. (4) Part 4: (d) Interpret your numerical result from part (c) in the physical context of the situation. What does the sign and magnitude of your answer tell you about the formation of the sand pile?

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