Aclosed rectangular container with a square base is to have a volume of 686 in3. the material for the top and bottom of the container will cost $4 per in2, and the material for the sides will cost $2 per in2. find the dimensions of the container of least cost.

Length = Width = 7 inches

Height = 14 inches

Step-by-step explanation:

Let's call  

x = length and width of the bottom and top

y = heigth

The bottom and top are squares, so their total area is

the area of one side is xy. As we have 4 sides, the total area of the sides is

The cost for the bottom and top would be

The cost for the sides would be

The total cost to made the box is

But the volume of the box is  

isolating

Replacing this expression in the formula of the total cost

The minimum of the cost should be attained at the point x were the derivative of the cost is zero.

Taking the derivative

If C' = 0 then

We have to check that this is actually a minimum.

To check this out, we take the second derivative

Evaluating this expression in x=7, we get C''>0, so x it is a minimum.

Now that we have x=7, we replace it on the equation of the volume to get

The dimensions of the most economical box are

Length = Width = 7 inches

Height = 14 inches

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