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# A box of constant volume c is to be twice as long as it is wide. The material on the top and four sides cost three times as much per square metre as that in the bottom.What are the most economic dimensions?

Arun Kumar IIT Delhi
6 years ago
Hi
$\\V=c \\l=2a \\b=a \\h=h \\2a^2h=c \\2(2ah {da \over dh}+a^2)=0 \\=>{da \over dh}={-a \over 2h} \\cost=3(lb+2lh+2hb)+lb \\=>cost=4lb+6lh+6hb \\=>cost=8a^2+12ah+6ah \\=>{d(cost) \over dh}={d(8a^2+18ah) \over dh}=16a{da \over dh}+18h{da \over dh}+18a=0 \\=>{da \over dh}={-18a \over (16a+18h)}={-a \over 2h} \\=>-36ah=-16a^2-18ah \\=>16a^2=18ah \\=>h={8 \over 9}a$

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty