**A rectangular storage container with an open top is to have a volume of 10 m3. The length of its 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.**

**Answer:**

V= base *height = 10

Heavy action on the width!

base =length *width

length = l = 2w

base= b in m²= l * w = 2w *w = 2w²
perimeter (distance around base) =p= 2(w +l) = 2( w+2w); 2(3w)= 6w
height = 10/b =10/2w² = 5/w²
sides= h*p=side in m²= 5/w² * 6w = 30w/ w²= 30/w

Cost, C = $10(base) +$6(sides)

C = 10(2w²) + 6(30/w)

C= 20w² +180/w

C= 20w² +180 w⁻¹

To Minimize Cost, get C’ by setting to zero and diffrentiate

0 = 40w + - 180 w⁻²

180/w² =40w

180/40 = w³

4.5 = w³

w = cube root 4.5 = 1.651

l= 2w = 2(1.651) =3.302

b= 2w or lw= 5.4514

sides= 30/w = 18.171

so cheapest cost: $10(5.4514) + $6(18.171)= 54.52 + 109.03= $163.55