A rectangular box with no top must have a volume of 2,500 cm³.

Three different materials will be used in the construction of this box. The bottom will be made out of material that costs 5 cents/cm², the front and back will be made out of material that costs 4 cents/cm², and material for the two sides costs 2 cents/cm².

What are the dimensions of the box (with volume 2,500 cm³) that will minimize the total cost of the materials?

let the dimension be x,y,z.  With z the hight, x the length of the front and back and y the length of the sides. then we have the volume as xyz=2500 and the cost as 5xy+4(2x+y)z  solving for z in the volume and substituting into the cost equation we get

5xy+10000(2x+y)/(xy)

differentiating by both x and y and setting each equation

equal to zero we get

5x^2y^2-10000(2x+y)+20000x=0 and

5x^2y^2-10000(2x+y)+10000y=0

subtracting these 2 equations we get

y=2x

putting this into the first equation we get

20x^4=20000x

x^3=1000=10^3

x=10

y=2*x=20

z=2500/(xy)=2500/200=12.5

so we want the dimensions to be 10x20x12.5 and this gives a total cost of $30

Posted by Daniel on 2009-05-15 13:11:18