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Question

A closed (cuboid) box with a square base is to have a volume 2000c.c. The material for the top and bottom of the box is to cost Rs.3 per square cm and the material for the sides is to cost Rs.1.50 per square cm. If the cost of the materials is to be the least, find the dimensions of the box.

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Solution

Let x be the length of side of square base.
y be height of cuboid and c be the total cost (in Rs.)
Area of bottom =x2
Area of top =x2
Combined area of top and bottom =2x2
Area of 4 side=4xy
cost of the material for top and bottom =3(2x2)
=6x2
Cost of the material for the side =1.5(4xy)=6xy
Total cost c=6x2+6xy ......(1)
Volume of the cost v= area depth =x2y
(i.e.,) x2y=2000 ....(given)
Eliminating from (1) and (2)
c(x)=6x2+12000x
c(x)=12x12000x2
12x212000=012(x2102)=0
x=10 or x2+10x+100=0
x2+10x+100=10 is not possible.
the critical number is x=10
c′′(x)=12+24000x3 c′′(10)=240001000>0
c is min at (10,1800)
Base length is 10 m and depth is 20 m.

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