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Grade 12Differential Calculus

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?

taniska

12 Years agoGrade 12

1 Answer

Arun Kumar

12 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

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