A covered box of volume 72cm3 and the base sides in a ratio of 1:2 is to be made. The length of all sides so that the total surface area is the least possible is
A
6,3,4
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B
8,4,3
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C
2,4,9
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D
3,6,2
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Solution
The correct option is A6,3,4 Let l,b,h be the dimension, l=2b, so V=lbh=2b2h ⇒72=2b2h ⇒h=36b2
The surface area S=2(lb+bh+hl)=2(2b2+b(36b2)+2b×36b2)=2(2b2+108b)=4(b2+54b) dSdb=4(2b−54b2),dSdb is zero if b=3 and d2Sdb2=4(2+108b3)>0
Hence, S is minimum when b=3.
So, the dimensions are 6,3,369=6,3,4