Answer:
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:
- rectangular box
- square base
- volumen
- base material cost
- sides material cost
- top material cost [/tex]
<u>You may find attached a draw with the dimentions of the box.</u>
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