Question

A rectangular storage container with an open top is to have a volume of 45m^3. the length of its base its base is twice its width. Material for the base costs $5 per square meter and meterial for the sides costs $4 per square meter. Find the cost for the cheapest such container.

Answer 1

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