A factory machine produces cardboard cones with base radius $r$ cm and vertical height $h$ cm. The cone volume, $V$ cm$^3$, is given by $V = \frac{1}{3}\pi r^2 h$. For the cones made by these machines, $h + r = 18$.
(i)[1]
Show that this can be rewritten as $V = 6\pi r^2 - \frac{1}{3}\pi r^3$.
(ii)[4]
Find the non-zero value of $r$ at which $V$ is stationary, and show that this stationary value is a maximum.
(iii)[1]
Find the greatest volume of a cone that these machines can produce.
(c(iii))[1]
Find the maximum volume that can be made by these machines.
Worked solution & mark scheme
This 7-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Hence write $V=\frac13\pi r^2(18-r)=6\pi r^2-\frac13\pi r^3$” …