Let h = BC and r = AC
You get the equation:
The volume of a cone is then:
Max volume is given by the derivative of the volume and set to 0. You have to use a substitution so that you have the volume in terms of r only, or in terms of h only.
You will have a value of r or h, depending on what you did and you can substitute that in the first equation to get the value of the other unknown, after which, you should get the ratio asked for.
EDIT: Really silly mistakes