Good evening Cheryl! I am trying to decipher what they meant by external surface area, because I am not sure whether it counts the bottom surface or not. Apparently the solution that I obtained for not counting the bottom surface leads to zero height, which does not make sense.

I will next do the same question with the bottom surface counted.

The reason why I did not count the bottom surface is that in general, the model of venues (such as concert halls) are usually being stuck on a platform (like for example models of HDB flats which are placed above some material) and thus the bottom surfaces should not be counted.

Good evening Cheryl! Here are my workings for this question, this time with the base included. It looks good this time.

Good evening Cheryl! For my second piece of workings with the base included, here is a more efficient way to obtain the corresponding value of h.

Also, it’s easy to prove that our value of r leads to a minimum total external surface area.

Since dA/dr = -2k/r^2 + 10#r/3,
d2A/dr2 = 4k/r^3 + 10#/3

which is clearly positive for positive values of r, and thus, d2A/dr2 > 0 implying a minimum value for the total external surface area.