Question

Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 30 cm

Answer 1

Answer:

$\frac{1}{9\pi} cm/s \ or \ 0.0354cm/s$

Step-by-step explanation:

question tests our knowledge of differentiation:

$\frac{dV}{dt} =100\frac{cm^3}{s}\\d=2r=30\\=>r=15$

Find $\frac{dr}{dt}$

Given the formula for a sphere's volume is

$V=\frac{4}{3}\pi r^3$

Take derivatives on both sides:

$\frac{d}{dt}(V)=\frac{d}{dt}(\frac{4}{3}\pr r^3)}\\\frac{dV}{dt}=\frac{4}{3}\pi (3r^2)\frac{dr}{dt}=4\pi r^2 \frac{dr}{dt}$

Hence;

$=\frac{dr}{dt}=\frac{1}{4\pi r^2}.\frac{dV}{dt}\\=\frac{dr}{dt}=\frac{1}{4\pi(15^2)}.100\\=\frac{1}{9\pi}$

Hence, the balloon radius increases at the rate of

$\frac{1}{9\pi} cm/s \ or \ 0.0354cm/s$