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

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.