The base of a rectangular box is to be twice as long as it is wide. The volume o

Question

The base of a rectangular box is to be twice as long as it is wide. The volume of the box is 256 cubic inches. The material for the top costs $0. 10 per square inch and the material for the sides and bottom costs $0.05 per square inch. Find the dimensions that will make the cost a minimum, length in width in height in Suppose that a company needs 1,200,000 items during a year and that preparation for each production run costs $750. Suppose also that it costs $27 to produce each item and $2 per year to store an item. Use the inventory cost model to find the number of items in each production run so that the total costs of production and storage are minimized. items/run

Explanation / Answer

1)let length of base be x , width be y , height be z

volume of box v =xyz =256

length is twice the width

=>x =2y

2y yz =256

=>z =128/y2

cost of top of box =0.10*x*y =0.1*2y*y=0.2y2

cost of sides of box=0.05[2*x*z +2*y*z]=0.05[2*2y*128/y2 +2*y*128/y2]=38.4/y

cost of bottom of box =0.05*x*y =0.05*2y*y=0.1y2

total cost of box C=0.2y2+38.4/y  +0.1y2

C=0.3y2+(38.4/y)

for minimum cost

dC/dy =0

0.6y+(-38.4/y2) =0

=>y3=38.4/0.6

=>y3 =64

=>y =4in

x =2y

=>x =8

z =128/42

=>z =8

length of rectangle =8in , width =4in,height =8in are the dimensions for minimum cost

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