A good place to start might be with the formulas for the volume V and surface area SA of a cylinder with height h and radius r. Also, it would probably be useful to note that you have been given that V = 500, 6<h<15, and that you are trying to minimize SA. So:

i) What is the formula for the volume V of a cylinder with height h and radius r?

ii) Noting that V = 500, solve for r in terms of h. (You have bounds on h, so you know you need to be working with h.)

iii) What is the formula for the surface area SA of a cylinder with height h and radius r?

iv) Into the formula in (iii), substitute for r using the results of (ii). You should now have SA in terms only of h.

v) Differentiate with respect to h, and minimize. Remember to check the endpoints of the interval for absolute minima.

vi) Back-solve for the radius, and clearly state the cylinder's dimensions.

If you get stuck, please reply showing your steps and reasoning so far.

Thank you!