A spherical balloon contains an ideal gas

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

Problem

A spherical balloon contains an ideal gas. The pressure

P

(in kilopascals) of the gas satisfies

P=\dfrac{kT}{V},

where

k

is a positive constant,

T

is the absolute temperature (in Kelvin), and

V

is the volume of the balloon (in cubic centimeters). The temperature and radius of the balloon both vary with time

t

(in seconds) according to

T(t)=300+2t, \qquad r(t)=10+0.1t^{2}, \qquad t\ge 0.

(The volume of a sphere is

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

.) All functions are differentiable. (a) Express

P

explicitly as a function of

t

by eliminating

T

and

V

. No derivatives are required in this part. (b) Derive an exact expression for

\dfrac{dP}{dt}

, and then evaluate

\dfrac{dP}{dt}

t=5

seconds. Include correct units. (c) At

t=0

the pressure is

P_{0}

. Use your result from part (b) to determine whether the pressure will have doubled by

t=5

seconds. Justify your conclusion quantitatively.