Question

You need to construct an open-top rectangular box with a square base that must hold a volume of exactly 475 cm3. The material for the base of the box costs 8 cents/cm2 and the material for the sides of the box costs 6 cents/cm2. The dimensions for a box that will minimize the cost of the materials used to construct box are:

Answer 1

Answer:

The dimensions of the box are:

x =  8,93 cm       and     h  =   5,95 cm

C(min) =  850,69 cents

Step-by-step explanation:

The volume of the box is:

V = x²*h          where    x is the side of the square base  and h the height

then    h  =  V/ x²  ⇒    h = 475 / x²

The total cost of box C is:

C  = C₁  +  4*C₂      Where C₁  and C₂  are the costs of the base and one lateral side respectevily

Then cost C =  8*x²   + 4* 6*h*x

The cost C as a function of x is

C(x)  =  8*x²  + (24* 475 /x² )*x

C(x)  =  8*x²  +  11400/x

Tacking derivatives on both sides of the equation

C´(x)  =  16*x -  11400/x²

C´(x)  =  0     ⇒    16*x  -  11400/x²  = 0

16*x³  =  11400     ⇒   x³  =  11400/16

x³ =  712,5

x  =  8,93  cm

and    h   =  475 / (8,93)²      ⇒      h  =  5,95  cm

C(min)  =  8*79,77  +  4* ( 8,93)*5,95

C(min)  =  638,16  +  212,53

C(min)  =  850,69 cents

To check if value x = 8,93 would make C(x) minimum we go to the second derivatives

C´´(x) =  16  +  22800/x³ > 0

Then we have a minimum of C at  x = 8,93