The volume of a cylinder, of radius r and height h, is given by \(\displaystyle \pi r^2h\) and its surface area is the circumference, \(\displaystyle 2\pi r\) times its height, \(\displaystyle 2\pi rh\). Here we should include the bottom, a disk of radius r, \(\displaystyle pi r^2\) (but not the top since the top is the hemisphere).
The volume of a sphere, of radius r, is \(\displaystyle (4/3)\pi r^3\) and its surface area is \(\displaystyle 4\pi r^2\) so the volume of the hemispherical top is \(\displaystyle (2/3)\pi r^3\) and the surface area is \(\displaystyle 2\pi r^2\).
So the total volume of this tank is \(\displaystyle \pi r^2h+ (2/3)\pi r^3\) and the surface area is \(\displaystyle 2\pi rh+ 3\pi r^2\). Now, what methods do you know for finding maxima and minima of such functions- like setting the partial derivatives equal to 0?
