Rectangle pool

Determine dimensions of open pool with a square bottom with a capacity 32 m3 to have painted/bricked walls with least amount of material.

Correct result:

a =  4 m
c =  2 m

Solution:

$V=32 \ \\ a=b \ \\ V=a^2 \ c \ \\ c=32/a^2 \ \\ S=a^2 +4ac=a^2 +4a (32/a^2) \ \\ S'=2a-128/a^2 \ \\ S'=0 \ \\ 2-128/a^3=0 \ \\ 1-64/a^3=0 \ \\ a^3=64 \ \\ a=\sqrt[3]{ 64 }=4 \ \\ b=a=4 \ \text{m}$

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