A rectangular storage container with an open top is to have a volume of 26 cubic meters. The length of its base is twice the width. Material for the base costs 13 dollars per square meter. Material for the sides costs 7 dollars per square meter. Find the cost of materials for the cheapest such container.

Question

cwrw238 · Answer

area of base = l*w = 2w*w = 2w^2
cost of base = 13*2w^2 = 26w^2
area of sides = 2*2wh + 2*wh  = 6wh
cost of the sides = 7*6wh = 42wh


total cost   C = 26w^2 + 42wh

cwrw238 · Answer

volume 26 = 2w^2h
h = 26/2w^2 = 13/w^2

substitute for h into the formula for C:

C  =  26w^2 + 42w * 13/w^2

C =  26w^2 +  546/w

cwrw238 · Answer

we need to find  the minimum value of this function

cwrw238 · Answer

now differentiate C with respect to w and equate to zero
solve for w 
then plug in this value for w into the formula for C and you have it.