A right circular cone has radius and height that may both vary with time

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

Problem

A right circular cone has radius

r(t)

and height

h(t)

that may both vary with time

t

. Its volume is given by

V(t)=\dfrac{1}{3}\pi r(t)^{2}h(t)

. (a) Describe, without performing any calculations, why the instantaneous rate of change

\dfrac{dV}{dt}

generally depends on both

\dfrac{dr}{dt}

and

\dfrac{dh}{dt}

even if only one of the dimensions is being directly altered by an external agent. (b) Suppose that at a particular instant the height is held constant while the radius is increasing. Explain the sign (positive, negative, or zero) of

\dfrac{dV}{dt}

at that instant, and justify your answer in the context of the formula for

V

. (c) In a different scenario the radius is being compressed (decreasing) while the height is simultaneously extended (increasing). Describe qualitatively how you would determine whether the volume is increasing or decreasing at that instant without explicitly computing

\dfrac{dV}{dt}

. Clearly state the factors you would consider.

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