Question

The volume of a sphere is increasing at a rate of $3$ cubic centimetres per second. How fast is the surface area increasing when the radius is $2$ centimetres ?

• A. $3c{m}^2/s$
• B. $6c{m}^2/s$
• C. $12c{m}^2/s$
• D. $1c{m}^2/s$

Answer 1

The correct option is A $3c{m}^2/s$

Volume of the sphere
$V=\frac{4}{3}\pi r^3\Rightarrow \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\Rightarrow 3=4\pi r^2\frac{dr}{dt}\Rightarrow \frac{dr}{dt}=\frac{3}{4\pi r^2}....\left(i\right)$
Surface area of the sphere
$S=4\pi r^2\Rightarrow \frac{dS}{dt}=8\pi r\frac{dr}{dt}\Rightarrow \frac{dS}{dt}=8\pi r\times \frac{3}{4\pi r^2}\Rightarrow \frac{dS}{dt}=\frac{6}{r}=3c{m}^2/s$