Find the surface area and dimensions of an open top rectangular prism with a constant volume of 9000 and height of 8

Question

open top rectangular prism, height of 8 and volume of 9000, find minimum surface area and dimensions.

Answer 1

Surface area S = xy + 16x + 16y; 8xy = 9000; y = 1125/x

S = 1125 + 16x + 18000/x; S' = 16 - 18000/x² = 0; x² = 1125; x = 15√5; y = 1125/(15√5) = 15√5

S'' = 36000/x³; S''(15√5) = 36000/(15√5)³ > 0. So, we have minimum surface if x = y = 15√5

S_min = 1125 + 480√5 = 2198.31