Rectangular box with a square base is to have a volume of 20 cubic feet. the material for the base costs 30 cents/square foot. the material for the sides costs 10 cents/square foot. the material for the top costs 20 cents/square foot. determine the dimensions that will yield minimum cost. let x = length of the side of the base.

The dimensions for the minimum cost are:

x = 2 feet

l = 5 feet (l is the hight of the box)

Step-by-step explanation:

Let´s find the data given:

You may find attached a draw with the dimentions of the box.

To know the box volume (V) we have to multiply base sides (x) and the box hight (l) as shown below:

(1)   V = x . x . l

And to determine the total cost, it´s necessary to add base, top and the 4 sides costs. Each one of then can be calculated as the material cost multiplied by the area. Then, total cost C is:

(2)  

Replacing square areas for base and top and rectangular area for sides:

(3)  

Let´s clear l from equation (1) just to replace it in equation (3):

/

(4)  

Replacing l found in (3):

(5)  

To determine an equation minimum is necessary to equal its derivative to zero. So, we have to find equation (5) derivative ():

ft.cents=0[/tex]

Finally with x value found above, we can obtain l from equation (4):

So, dimensions for the minimum cost are:

x = 2 feet

l = 5 feet