I have a question in my Calculus 1 homework that I'm not sure where to begin with.
I need to calculate the instantaneous rate of change of the volume of a cylinder as the radius varies while the surface area is held fixed.
I know that volume $V=\pi r^2 h$ and surface area $S=2\pi rh+2\pir^2$ however I'm not sure how to relate them in an equation.
Thanks for your help in advance!
Best Answer
Hint:
Find $h$ from the equation of the surface: $$ h=\frac{S}{2\pi r}-r $$ and substitute in the volume: $$ V=\pi r^2\left(\frac{S}{2\pi r}-r \right) $$ This is the equation that gives the volume as a function o f $r$ for a given total surface $S$.