Aclosed box with a square base is to have a volume of 16,000 cm^3. the material for the top and bottom of the box costs $3 per square centimeter, while the material for the sides costs $1.50 per square centimeter. find the dimensions of the box that will lead to the minimum total cost. what is the minimum total cost ?

base side = 20cm

height = 40 cm

minimum cost = $7200

Step-by-step explanation:

Let x (cm) be the base side and h (cm) be the height of the box. Since the volume of the box must be 16000 cubic cm, we have:

The areas of the top and bottom is

And so their cost is:

The areas of the sides is

And so their cost is:

We can substitute into the equation above to get

So the total cost of the bases and the sides:

This is the function we would want to minimize. We can start by taking the 1st derivative, and set it to 0

So the height would be

So the minimum total cost would be

step-by-step explanation:

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