Air is pumped into a spherical balloon at a constant rate of

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

Problem

0.040\text{ m}^{3}\text{ s}^{-1}

. Let

r

be the radius (in meters) of the balloon,

V

its volume (in cubic meters), and

S

its surface area (in square meters). The formulas

V=\dfrac{4}{3}\pi r^{3}

and

S=4\pi r^{2}

apply. (a) Derive an expression for the rate of change

\dfrac{dS}{dt}

of the surface area in terms of the volume–inflation rate

\dfrac{dV}{dt}

and the instantaneous radius

r

. (b) The balloon material can safely withstand surface–area expansion rates up to

6.0\text{ m}^{2}\text{ s}^{-1}

. Using your result from part (a) and the given pumping rate, determine the radius (to the nearest centimeter) at which this safety limit is reached. (c) Explain, in context, what your answer in part (b) implies about the continued inflation of the balloon once its radius exceeds the computed value.

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