Mathematics

Problem

A spherical balloon is deflating so that its radius decreases at a constant rate. Which statement about the balloon’s volume

V

is most accurate? a)

\displaystyle \frac{dV}{dt}

is constant because

\dfrac{dr}{dt}

is constant. b)

\displaystyle \frac{dV}{dt}

is approximately zero since the surface area

4\pi r^{2}

shrinks as the balloon deflates. c)

\displaystyle \frac{dV}{dt}

is negative because the volume is decreasing. d)

\displaystyle \frac{dV}{dt}

is positive because the cubic relationship

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

turns the negative rate of

r

into a positive rate for

V