optimzation

How can we explain why you were wrong when you have not even given us an outline of your working ?

h =  300/ (pi r^2)    →  h^2  =  (90000/ pi^2)r^-4

Subbing   x for  r   we have  that

S   =  (pi * x)  ( x^2  +  (90000/pi^2)x^-4 )^1/2

The Calculus is messy  but this graph  shows  that  there is  no  x = r   that will  produce a max.....only  a  min

https://www.desmos.com/calculator/zpyqlutuoq

A radius of  ≈  4.07 cm    will produce a minimum surface area  of  ≈ 90.229 cm^2

h =  300 / ( pi * 4.07²)  ≈  5.76  cm