Question Number 98278 by liki last updated on 12/Jun/20
Answered by mr W last updated on 12/Jun/20
$${side}\:{length}\:{of}\:{square}\:={a} \\ $$$${height}\:{of}\:{box}\:={h} \\ $$$${volume}\:{V}={a}^{\mathrm{2}} {h} \\ $$$${surface}\:{S}={a}^{\mathrm{2}} +\mathrm{4}{ah}={a}^{\mathrm{2}} +\mathrm{4}{a}\frac{{V}}{{a}^{\mathrm{2}} }={a}^{\mathrm{2}} +\frac{\mathrm{4}{V}}{{a}} \\ $$$$\frac{{dS}}{{da}}=\mathrm{2}{a}−\frac{\mathrm{4}{V}}{{a}^{\mathrm{2}} }=\mathrm{0} \\ $$$$\Rightarrow{a}^{\mathrm{3}} =\mathrm{2}{V} \\ $$$$\Rightarrow{a}=\sqrt[{\mathrm{3}}]{\mathrm{2}{V}} \\ $$$${h}=\frac{{V}}{{a}^{\mathrm{2}} }=\frac{{V}}{\sqrt[{\mathrm{3}}]{\mathrm{4}{V}^{\mathrm{2}} }}=\frac{\sqrt[{\mathrm{3}}]{{V}}}{\sqrt[{\mathrm{3}}]{\mathrm{4}}} \\ $$$$\Rightarrow\frac{{h}}{{a}}=\frac{\mathrm{1}}{\sqrt[{\mathrm{3}}]{\mathrm{4}}×\sqrt[{\mathrm{3}}]{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by liki last updated on 12/Jun/20
$$..\boldsymbol{\mathrm{blessed}}\:\boldsymbol{\mathrm{sir}},\mathrm{i}\:\boldsymbol{\mathrm{appriciate}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{time}} \\ $$