In a right circular conical tank with a small hole at the vertex

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

Problem

In a right circular conical tank with a small hole at the vertex, students model the draining process by expressing the radius

r

of the water surface as a function of the water depth

h

before differentiating the volume

V

with respect to time

t

. Which of the following best explains the primary advantage of substituting

r=f(h)

before differentiating? (a) It avoids the need to apply the chain rule because

V

becomes a direct function of

h

. (b) It shows that

\displaystyle \frac{dr}{dt}=\frac{dh}{dt}

, so only one rate must be considered when differentiating. (c) It eliminates one dependent variable, letting

\displaystyle \frac{dV}{dt}

be written solely in terms of

h

and \,

\displaystyle\frac{dh}{dt}

. (d) It permits the constant

\pi

to be dropped from the volume formula, streamlining the computation.