When both the radius and the height of a right circular cone change with respect to time

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

When both the radius

r

and the height

h

of a right circular cone change with respect to time

t

, which of the following best explains why

\dfrac{dV}{dt}

cannot be found by substituting

\dfrac{dr}{dt}

\dfrac{dh}{dt}

into a single–variable derivative of

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

? (a) Because

r

and

h

are geometrically linked in a cone, making one variable determine the other; therefore differentiating with respect to a single variable is sufficient. (b) Because differentiating

V

with respect to

r

treats

h

as constant (and vice-versa), which is invalid when both are changing, so using only one rate omits part of the change in volume. (c) Because

V

depends on two independent variables, so each variable’s rate of change contributes separately to

\dfrac{dV}{dt}

. (d) Because using

\dfrac{dV}{dr}

\dfrac{dV}{dh}

alone leaves the result without the required units of volume per time, revealing that a single-rate substitution is incorrect.

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