Question

A rectangular storage container with an open top is to have a volume of 20 cubic meters. The length of its base is twice the width. Material for the base costs 20 dollars per square meter. Material for the sides costs 7 dollars per square meter. Find the cost of materials for the cheapest such container.

Answer 1

V = 20 = l*w*h
l = 2w
V = 20 = (2w)*w*h = 2w^2 * h
h = 10/(w^2)

Side A: l *h = (2w)(10/(w^2)) = 20/w
Side B: w*h = w*(10/(w^2)) = 10/w
Base = l*w = 2w*w = 2w^2

Total Area of Side A: 2 * (20/w) = 40/w
Total Area of Side B: 2 * (10/w) = 20/w

Total Area = (40/w)+(20/w) + 2w^2 = 60/w + 2w^2

Total area weighted cost = 7*(60/w) + 20*(2w^2) = 420/w + 40w^2

Find the critical point: 1. Take derivative. 2. Set to zero. 3. Solve for w.

Derivative = 80w - (420/(w^2)) = 0
80w = 420/w^2 --> 80w^3 = 420
w^3 = 420/80
w = apprx (1.738)

Re-input into cost formula: 420/1.738 + 40*(1.738)^2 = apprx (362.48)

Answer 2

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