You have asked to design a rectangle box with a square base and an open top. The volume of the box must be620 cm to the 3rd power. the cost of the material for the base is $0.40 per square cm and the cost of the material for the side is $0.10 per square cm. How to determine the dimension of the box that will minimize the cost of manufacturing. What is the minimum cost? in dollars and rounded to the nearest cent.

$69.21

Step-by-step explanation:

Since the box has a square base the length and breadth of the box will be equal. Let it be

Let h be the height of the box

V = Volume of the box =

Now surface area of the box with an open top is given

Differentiating with respect to x we get

Equating with zero

Double derivative of the function

So, x at 10.74 is the minimum value of the function.

So, minimum length and breadth of the box is 10.74 cm while the height of the box is 5.37 cm.

The total area of the sides is

The area of the base is

Cost of the base is $0.40 per square cm

Cost of the side is  $0.10 per square cm

Minimum cost would be

The minimum cost of the box is 69.21 dollars.

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