Answer:
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
