# Maximum volume of a cone.

I'm machining a component as a means of testing one of our companies new machines. The objective is to manufacture a cone of maximum possible volume. The volume of a cone is given by: pir2h/3.

Given that h = 6 - r, how am i to calculate the maximum possible volume by means of integration.

Many thanks, Joe.

You can't "by means of integration! You can "by means of differentiation". Putting h= 6- r in $\pi r^2h/3$ gives $\pi r^2(6-r)/3= \pi/3(6r^2- r^3)$. The derivative of that is $\pi/3(12r- 3r^2)= \pi(4r- r^2)$. The maximum possible volume is given by the (non-zero) r value that makes that 0.