Question

uppose we want to build a rectangular storage container with open top whose volume is $$ cubic meters. Assume that the cost of materials for the base is$$ dollars per square meter, and the cost of materials for the sides is $$ dollars per square meter. The height of the box is three times the width of the base

Answer 1

Answer:

a = length of the base = 2.172 m

b = width of the base = 1.357 m

c = height = 4.072 m

Step-by-step explanation:

Suppose we want to build a rectangular storage container with open top whose volume is 12 cubic meters. Assume that the cost of materials for the base is 12 dollars per square meter, and the cost of materials for the sides is 8 dollars per square meter. The height of the box is three times the width of the base. What’s the least amount of money we can spend to build such a container?

lets call a = length of the base

b = width of the base

c = height

V = a.b.c = 12

Area without the top:

Area = ab + 2bc + 2ac

Cost  = 12ab + 8.2bc + 8.2ac

Cost = 12ab + 16bc + 16ac

height = 3.width

c = 3b

Cost = 12ab + 16b.3b + 16a.3b = 12ab + 48b² + 48ab = 48b² + 60ab

abc = 12 → ab.3b = 12 → 3ab² = 12 → ab² = 4 → a = 4/b²

Cost = 48b² + 60ab = 48b² + 60b.4/b² = 48b² + 240/b

C(b) = 48b² + 240/b

C'(b) = 96b - 240/b²

Minimum cost: C'(b) = 0

96b - 240/b² = 0

(96b³ - 240)/b² = 0

96b³ - 240 = 0

96b³ = 240

b³ = 240/96

b³ = 2.5

b = 1.357m

c = 3b = 3*1.357 = 4.072m

a = 4/b² = 2.172m