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# A box of constant volumec is to be twice as long is it is wide. The material on the top and four sides cost three times as much per square metre as thatin the bottom. What are the most economical dimensions?[3c16]1/3,2[3c16]1/3 and [32c81]1/3.[9c16]1/3,2[9c16]1/3 and [32c81]1/3.[9c16]1/3,2[9c16]1/3 and [32c9]1/3.[81c16]1/3,2[81c16]1/3 and [32c81]1/3.

A
[9c16]1/3,2[9c16]1/3 and [32c9]1/3.
B
[3c16]1/3,2[3c16]1/3 and [32c81]1/3.
C
[9c16]1/3,2[9c16]1/3 and [32c81]1/3.
D
[81c16]1/3,2[81c16]1/3 and [32c81]1/3.
Solution
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#### Let the breadth be x, length be 2x and height be hV=x.2x.h⇒c=2x2h ....(1)Area of bottom =2x2 =Area of topArea of sides =2xh+2xh+xh+xh=6xh.If R rupees be the cost of material for bottom then for the top and sides is 3R.∴E=R(2x2)+3R(2x2+6xh)⇒E=R(8x2+18xh)or E=R(8x2+18xc2x2)⇒E=R(8x2+9cx)where R and c are constants.dEdx=R(16x−9cx2)For maximum or minimum, dEdx=0∴x=(9c16)1/3Also, d2Edx2=R(16+18cx3)=+iveand hence minimum.∴ dimensions are[9c16]1/3,2[9c16]1/3 and [32c81]1/3.

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