An open-top rectangular tank with a square base and a volume of 32 ft3 is to be built. what dimensions minimize the amount of material required to build this tank? show that your result is a minimum.

x  =  8  ft

h =  1/2  ft  

Step-by-step explanation:

Let  x be side of the base then area of the base is  x²

Let h be the height of the tank    

Tank volume is   32 ft³     and is   32  =  x²*h    then  h  = 32 /x²

Area of base  + lateral area = total area (A)

A = x²  + 4*x*h      ⇒   A = x²  + 4*x*(32/x²)          A = x²  + 128/x

A(x)  = x²  + 128/x     (1)

Taking derivatives on both sides of the equation

A´(x)   =  2x   -  128/x²             A´(x)   =  0             2x   -  128/x²   =  0

(2x² -128) / x²    =  0

2x²  -  128  = 0

x²  =√64

x  =  8  ft

The result is minimum since replacing in equation (1)  x = 8 we get

A(x) > 0

And

h  =  32/x²

h =  1/2  ft  

1,8,15,22,29,36,43,50

step-by-step explanation:

a.) list the factors of 40.

1/2/4/5/8/10/20/40

b.) list the factors of 40 that represent the number of cans that could be on each shelf.

5

c.) how many different arrangements of cans are possible?

just one because if you put more on one shelf you will go way over 40 and the cabinet will probably break.

d.) what are the possible arrangements of the cans?

put 5 cans on each self to reach 40 cans in total.

e.) which arrangement uses all the shelves?

5 because you can put 5 cans on each shelf, and since there are 8 shelfs you are equally going to get 40 cans.