Abox is contructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $3 per square foot and the metal for the sides costs $10 per square foot. find the dimensions that minimize cost if the box has a volume of 15 cubic feet.

Let x ft be the length of the base square and y ft be the height of the box.

The volume of the box is

Since the box has a volume of 15 cubic feet, then

You need to construct two squares from the metal that cost $3 per square foot.

The area of each square is and the total cost for these two squares is

The area of each side face is Then the total cost for sides is

Let S(x) be the function that represents total cost of the box, then

Find the derivative:

When then

The dimensions of the box are:

length and width -

height -

x=1

y=4

Step-by-step explanation:

We know that the diagonals bisect each other so

3x+2 = y+1

and x+3 =y

Substituting x+3 for y in the first equation

3x+2 = (x+3) +1

3x+2 = x+4

Subtract x from each side

3x-x+2 =x-x+4

2x+2 = 4

Subtract 2 from each side

2x +2-2 =4-2

2x=2

Divide by 2

2x/2 = 2/2

x=1

Now we need to find y

and x+3 =y  We know x=1

1+3 =y

4=y