Question

An open box with a square base is to have a volume of 18 ft^{3}

Find the box dimensions that minimize the amount of material used. (Round your answers to two decimal places.)

Step 1

Consider the thickness of the material to be negligible.

Since the base is square shaped, the length and width of the box would be same as the side of the base.

Let the side of the base be *s* ft and height be *h* ft.

Given the volume of the box is 18ft^{3}.

Volume of the box is given by *s*^{2}*h*.

Hence,

Step 2

The material used is minimum when the surface area would be minimum.

The surface area of an open box is given by:

Plugging the value of *h*:

Step 3

A function has a maximum or minimum value at a point where the first derivative is 0.

Applying derivative for the functio...

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