A container with a square base and open top must have a volume of 64000 cm3.

Question

A container with a square base and open top must have a volume of 64000 cm3. The cost of material used to make the sides of the container is $5/cm2, while it is $10/cm2 for the base. Find the smallest possible total cost of material used to construct the container

Answer 1

Volume: S^2*h = 64000

surface area = S^2 + 4*S*h <-- no top

cost function: 10S^2 + 20*S*h <--- $10 per square centimeter for the base

$5 per square centimeter for the sides

substitutes h=64000/S^2 into the cost function

Cost(S) = 10S^2 + 20*S * 64000/S^2

= 10S^2 + 1280000/S

= 10S^2 + 1280000 S^(-1)

minimizing:

0= dCost/dS = 20S - 1280000 S^(-2)

multiplies both sides by S^2

0 = 20S^3 - 1280000

0 = S^3 - 64000

S^3 = 64000

S = 40

then 1600*h = 64000

h = 40

Height = 40 cm

Base has side measure of 40 cm as well

10*1600 + 20*40*40 = 16000+ 16000 = 32000 is the cost