A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
I understand the calculus for the most part, but creating the initial formula for this is confusing me to no end. So far I've got: Cost = 10(b^2) + 4(6b*h) where b=l*w where l=2w so b = 2w*w = 2w^2.
Now I assume I mess up somewhere when I go with 10m cubed = (l*w)*h. So 10 = b*h. b = 2w^2, So h = 10/2w^2. I proceed to throw that into the Cost equation so that C = 10(2w^2)^2 + 24(2w^2)(10/2w^2). Which simplifies to 10(2w^2)^2 + 240.
C' = 20(2w^2) * 4w = 10w^3
Not quite sure where I messed up, as the problem is kinda long. I would appreciate it if someone could point out my error so I could work on it. I have trouble with these optimization problems as they are mostly word problems that tend to confuse me.