The surface area of a spherical balloon and its radius satisfy for all

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

The surface area

A(t)

of a spherical balloon and its radius

r(t)

satisfy

A=4\pi r^{2}

for all

t

. At an instant when

r=5\text{ cm}

and the surface area is increasing at

\dfrac{dA}{dt}=30\text{ cm}^2\!\big/\!\text{s}

, which of the following equations correctly expresses

\dfrac{dr}{dt}

in terms of

\dfrac{dA}{dt}

and

r

? (a)

\displaystyle \frac{1}{4\pi r}\,\frac{dA}{dt}

(b)

\displaystyle 8\pi r\,\frac{dA}{dt}

(c)

\displaystyle \frac{1}{8\pi r}\,\frac{dA}{dt}

(d)

\displaystyle \frac{1}{4\pi r^{2}}\,\frac{dA}{dt}