A rectangular storage container with an open top is to have a volume of 5 m3.

Question

A rectangular storage container with an open top is to have a volume of 5 m³. The length of its base is twice the width. Material for the base costs $4 per square meter. Material for the sides costs $7 per square meter. Find the cost of materials for the cheapest such container. Approximate your answer to the nearest cent. 

Answer 1

The given volume provides the "constraint," which will allow us to write the height of the box in terms of its width.

We can then use the various prices and surface area to write an equation for cost in terms of the width only. Lastly, we will differentiate this equation, set it = 0, an solve to get the ideal width, which we then plug in to the cost function to find the minimum cost:

V = w·2w·h = 5 ; h = 5/2·w^-2

C(w) = 4·w·2w + 2·7·w·(5/2·w^-2) + 2·7·2w·(5/2·w^-2) = 8w² +105w^-1

C^'(w) = 16w - 105w^-2 = 0

16w³ - 105 = 0

w = ³√(105/16) ; C_min = 8·³√(105/16)² + 105 / ³√(105/16)