Arectangular box with a volume of 784ft^3 is to be constructed with a square base and top. the cost per square foot for the bottom is 20 cents, for the top is 16 cents, and for the sides is 1.5 cents. what dimensions will minimize the cost? the length of one side of the base is ? the height of the box is?

Let's dimension the horizontal length with the name X and the vertical dimensions as Y.

In this way the total volume will be given under the function

The cost for the bottom is given by

While the cost for performing the top by

The cost for performing the sides would be given by

Therefore the total cost would be

The total volume is equivalent to

Replacing in our cost function

Obtaining the first derivative and equalizing to zero we will obtain the ideal measure, therefore

Then,

In this way the measures of the base should be 6.3947ft (width and length) and the height of 19.17ft.

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e/c, where e is its total energy