Minimization:
Suppose that a manufacturer wants to make an ice cream cone that holds 8 cubic inches of ice cream when filled (the top of the cone is scraped to be level). What height and radius minimize the cost of producing the cone?

I've already established the SA and Volume of a Cone are 1/3pi*r^2*h and pi*r^2+pi*r*sqrt(r^2+h^2), respectively. Additionally, that h= 24/(pi*r^2). I'm trying to take the derivative of the surface area to find a value for the radius, but am running into troubles. Should I substitute h, then derive? Lastly, please derive Surface Area so I can check my work.

Question

Minimization:
Suppose that a manufacturer wants to make an ice cream cone that holds 8 cubic inches of ice cream when filled (the top of the cone is scraped to be level). What height and radius minimize the cost of producing the cone? 

I've already established the SA and Volume of a Cone are 1/3pi*r^2*h and pi*r^2+pi*r*sqrt(r^2+h^2), respectively. Additionally, that h= 24/(pi*r^2). I'm trying to take the derivative of the surface area to find a value for the radius, but am running into troubles. Should I substitute h, then derive? Lastly, please derive Surface Area so I can check my work.