# 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

#1

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.

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