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:

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

$c=32\mathrm{/}{a}^2=32\mathrm{/}{4}^2=2\text{ m}$

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