A thin layer of liquid ink spreads uniformly across a horizontal table

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

Problem

A thin layer of liquid ink spreads uniformly across a horizontal table, forming a circular film whose thickness remains exactly 0.50 mm. At noon the radius of the film is 4.0 cm and is increasing at a rate of 0.030 cm min⁻¹. Assume the density and thickness of the ink remain constant during the spreading process. (a) Express the volume

V

(in cubic centimeters) of the ink film as a function of its radius

r

(in centimeters). (b) Derive an expression for

\dfrac{dV}{dt}

in terms of

r

and

\dfrac{dr}{dt}

. Then calculate the instantaneous rate at which the volume is changing at noon. Include appropriate units. (c) Determine the instantaneous rate of change of the area of the film,

\dfrac{dA}{dt}

, at noon. Give numerical value and units. (d) Interpret the physical meaning of your numerical answer in part (b) in the context of the spreading ink.

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