A hot-air balloon’s envelope expands as it is heated

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

Problem

A hot-air balloon’s envelope expands as it is heated. The balloon’s volume (in m

^{3}

) at time

t

minutes after ignition is

V(t)=900+15t.

The air temperature inside the envelope (in K) is related to its volume by

T(V)=280+0.04V,

and the internal air pressure (in atm) is related to temperature by

P(T)=1.2-0.0005T.

Assume all three expressions remain valid for

0\le t\le 20

. (a) Derive a formula for

\dfrac{dP}{dt}

in terms of

\dfrac{dP}{dT}

\dfrac{dT}{dV}

, and

\dfrac{dV}{dt}

. (b) Compute

\dfrac{dP}{dt}

when

t=4

minutes. Show all work and state units for the final result. (c) The safe operating range for the balloon requires that the magnitude of the pressure change not exceed

0.015\text{ atm/min}

. Determine whether the balloon is within the safe range at

t=4

minutes and justify your conclusion.

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