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Question Number 74209 by FCB last updated on 20/Nov/19

Answered by mind is power last updated on 20/Nov/19

a^2 +b^2 +c^2 =48  a≥b≥c  ⇒48≥3a^2 ⇒a^2 ≤16⇒a≤4  a=4⇒b^2 +c^2 =32⇒(b,c)=(4,4)  a=3⇒b^2 +c^2 =39  non solution  a=2⇒b^2 +c^2 =44⇒non solution  a=1⇒b^2 +c^2 =47⇒non solution  a=0⇒b^2 +c^2 =48⇒non solution  ⇒(a,b,c)=(4,4,4)  our expression= 3.16.(√(144))=3.16.12=576

$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{48} \\ $$$${a}\geqslant{b}\geqslant{c} \\ $$$$\Rightarrow\mathrm{48}\geqslant\mathrm{3}{a}^{\mathrm{2}} \Rightarrow{a}^{\mathrm{2}} \leqslant\mathrm{16}\Rightarrow{a}\leqslant\mathrm{4} \\ $$$${a}=\mathrm{4}\Rightarrow{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{32}\Rightarrow\left({b},{c}\right)=\left(\mathrm{4},\mathrm{4}\right) \\ $$$${a}=\mathrm{3}\Rightarrow{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{39}\:\:{non}\:{solution} \\ $$$${a}=\mathrm{2}\Rightarrow{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{44}\Rightarrow{non}\:{solution} \\ $$$${a}=\mathrm{1}\Rightarrow{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{47}\Rightarrow{non}\:{solution} \\ $$$${a}=\mathrm{0}\Rightarrow{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{48}\Rightarrow{non}\:{solution} \\ $$$$\Rightarrow\left({a},{b},{c}\right)=\left(\mathrm{4},\mathrm{4},\mathrm{4}\right) \\ $$$${our}\:{expression}=\:\mathrm{3}.\mathrm{16}.\sqrt{\mathrm{144}}=\mathrm{3}.\mathrm{16}.\mathrm{12}=\mathrm{576} \\ $$

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