A small air bubble is rising slowly through seawater

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

Problem

A small air bubble is rising slowly through seawater. Because heat exchange with the surrounding water is rapid, you may assume the bubble’s temperature remains constant. Let

P(t)

be the absolute pressure in kilopascals acting on the bubble and

V(t)

its volume in cubic centimeters. Boyle’s law applies:

PV=k

, where

k

is constant. At one instant the pressure is

P=250\text{ kPa}

and is changing at a rate of

\dfrac{dP}{dt}=-9.8\text{ kPa\,s}^{-1}

. The bubble’s volume at that instant is

V=1.20\text{ cm}^{3}

. (a) Differentiate Boyle’s law with respect to time to obtain a relationship between

\dfrac{dP}{dt}

and

\dfrac{dV}{dt}

. Do not substitute numerical values yet. (b) Use your result from part (a) to calculate

\dfrac{dV}{dt}

at the given instant. Include correct units. (c) Explain, in the context of the physical situation, why the signs of

\dfrac{dP}{dt}

and

\dfrac{dV}{dt}

are different.

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