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Question Number 60422 by Tawa1 last updated on 20/May/19

Commented by MJS last updated on 20/May/19

what′s a “lidless box”? what are the “ends” of a box?

$$\mathrm{what}'\mathrm{s}\:\mathrm{a}\:``\mathrm{lidless}\:\mathrm{box}''? \\ $$$$\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:``\mathrm{ends}''\:\mathrm{of}\:\mathrm{a}\:\mathrm{box}? \\ $$

Commented by Tawa1 last updated on 20/May/19

I don′t know sir. that is the question.

$$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{know}\:\mathrm{sir}.\:\mathrm{that}\:\mathrm{is}\:\mathrm{the}\:\mathrm{question}. \\ $$

Commented by Tawa1 last updated on 20/May/19

Maybe question is wrong sir

$$\mathrm{Maybe}\:\mathrm{question}\:\mathrm{is}\:\mathrm{wrong}\:\mathrm{sir} \\ $$

Commented by MJS last updated on 20/May/19

if it′s a box with square sides and floor but without a cap the volume is a^3 =3.5 and the surface is 5a^2 5(((3.5))^(1/3) )^2 ≈11.53

$$\mathrm{if}\:\mathrm{it}'\mathrm{s}\:\mathrm{a}\:\mathrm{box}\:\mathrm{with}\:\mathrm{square}\:\mathrm{sides}\:\mathrm{and}\:\mathrm{floor}\:\mathrm{but} \\ $$$$\mathrm{without}\:\mathrm{a}\:\mathrm{cap}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{is}\:{a}^{\mathrm{3}} =\mathrm{3}.\mathrm{5}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{surface}\:\mathrm{is}\:\mathrm{5}{a}^{\mathrm{2}} \\ $$$$\mathrm{5}\left(\sqrt[{\mathrm{3}}]{\mathrm{3}.\mathrm{5}}\right)^{\mathrm{2}} \approx\mathrm{11}.\mathrm{53} \\ $$

Commented by Tawa1 last updated on 20/May/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Answered by mr W last updated on 21/May/19

box without lid box with square base b×b height of box h V=b^2 h ⇒h=(V/b^2 ) S=b^2 +4bh=b^2 +((4V)/b) (dS/db)=2b−((4V)/b^2 )=0 ⇒b=((2V))^(1/3) =((2×3.5))^(1/3) =(7)^(1/3) =1.913 m S_(min) =1.913^2 +((4×3.5)/(1.913))=10.978 m^2

$${box}\:{without}\:{lid} \\ $$$${box}\:{with}\:{square}\:{base}\:{b}×{b} \\ $$$${height}\:{of}\:{box}\:{h} \\ $$$${V}={b}^{\mathrm{2}} {h} \\ $$$$\Rightarrow{h}=\frac{{V}}{{b}^{\mathrm{2}} } \\ $$$${S}={b}^{\mathrm{2}} +\mathrm{4}{bh}={b}^{\mathrm{2}} +\frac{\mathrm{4}{V}}{{b}} \\ $$$$\frac{{dS}}{{db}}=\mathrm{2}{b}−\frac{\mathrm{4}{V}}{{b}^{\mathrm{2}} }=\mathrm{0} \\ $$$$\Rightarrow{b}=\sqrt[{\mathrm{3}}]{\mathrm{2}{V}}=\sqrt[{\mathrm{3}}]{\mathrm{2}×\mathrm{3}.\mathrm{5}}=\sqrt[{\mathrm{3}}]{\mathrm{7}}=\mathrm{1}.\mathrm{913}\:{m} \\ $$$${S}_{{min}} =\mathrm{1}.\mathrm{913}^{\mathrm{2}} +\frac{\mathrm{4}×\mathrm{3}.\mathrm{5}}{\mathrm{1}.\mathrm{913}}=\mathrm{10}.\mathrm{978}\:{m}^{\mathrm{2}} \\ $$

Commented by Tawa1 last updated on 21/May/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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