An open box is to be constructed so that the length of the base is 4 times larger than the width of the base. If the cost to construct the base is 5 dollars per square foot and the cost to construct the four sides is 3 dollars per square foot, determine the dimensions for a box to have volume = 25 cubic feet which would minimize the cost of construction.

Question

Values for dimension of the base are: ?

The height of the box is: ?

Answer 1

l=5w

costbase= lw*5
cost sides=3(2lh+2wh)

volume=lwh
25=lwh
or h= 25/wl

costtotal=5(lw*5)+3(2l*25/wl + 2w*25/wl)
costtotal=25lw+ 150/w+150/l

but l=5w
costtotal= 125w^2+150/w+30/w

take the derivative of cost.
costtotal'=0=250w-150/w^2-30/w^2

solve for w:

250w^3-180=0
w^3=180/250
w= cubroot 7.2

l= 4 cubroot 7.2

h= 25/(lw)

check my work.

Answer 2

moron