Consider a general right circular cone of height and base radius

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

Problem

Consider a general right circular cone of height

H

and base radius

R

. Suppose the cone is partially filled with a liquid to a depth

h

(where

0<h<H

). The liquid drains through an opening at the tip, and at a particular instant the volume of liquid is decreasing at the constant rate

k

(in units of volume per unit time). Let

r

be the radius of the liquid surface corresponding to depth

h

. (a) Explain why

\dfrac{r}{h}=\dfrac{R}{H}

for any instantaneous depth

h

. (b) Using part (a), express the volume

V

of the liquid solely in terms of

h

R

, and

H

, without resorting to numerical substitution. (c) By differentiating implicitly with respect to time, write a general formula for

\dfrac{dh}{dt}

in terms of

h

R

H

, and

k

. Do not substitute numerical values. (d) Describe how the magnitude of

\dfrac{dh}{dt}

changes as the liquid depth decreases toward zero, assuming

k

remains constant. Provide a conceptual explanation based on the geometry of the cone.