Volume of water = volume of cylinder + volume of hemisphere

= πr²(h - r) + ⅔πr³ = 144π

πr²h - πr³ + ⅔πr³ = 144π

πr²h = 144π + ⅓πr³

h = 144/r² + ⅓r = 144/r² + r/3

Surface area in contact with water, A

= Curved surface area of cylinder + curved surface area of hemisphere

= 2πr(h-r) + 2πr²

= 2πrh - 2πr² + 2πr²

= 2πrh

= 2πr(144/r² + r/3)

= 288π/r + ⅔πr²

dA/dr = -288π/r² + 4/3 πr


when dA/dr = 0,

4/3 πr - 288/r² = 0

4/3 πr = 288/r²

4/3 πr³ = 288

r³ = 216/π

r = ³√(216/π) = 6/³√π

h = 144/(6/³√π)² + (6/³√π) / 3

h = 4 ³√π² + 2 / ³√π


d²A/dr² = 576π/r³ + 4/3 π

= 576π / (216/π) + 4/3 π

= 8/3 π² + 4/3 π > 0

(minimum)

Thanks for spotting out my mistake :) Did not read the question carefully :)

Thanks!