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A box with a square base and open top is to have a volume of 72000 cubic inches. the material for the base costs 4 cents per square inch, while the material for the sides costs 6 cents per square inch. find the dimensions which minimize the cost of material.

Respuesta :

Volume of box=a²h=72000, where a is the side of the base and h the height. h=72000/a²
Area of base=a².
Area of sides=4ah. Total=a²+4ah.
Total cost=4a²+24ah=4a²+24×72000/a=4a²+1728000/a.
Differentiate wrt a: 8a-1728000/a². When 8a=1728000/a², there is a minimum.
So 8a³=1728000, a³=216000, a=60.
Therefore h=72000/3600=20.
The dimensions are: base side=60 in, height=20 inches.

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