Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 120 in. Find the dimensions of a rectangular package that has a square cross section and largest volume that may be sent by priority mail. (Hint: The length plus the girth is 4x l.)

x = 20 in

L = 40 in

Step-by-step explanation:

Solution:-

- Denote the following:

 The side of square cross section = x

 The length of package = L

- Given that the combined length "L" of the package and girth "P" of the package must be less than and equal to 120 in

- The girth of the package denotes the Perimeter of cross section i.e square:

                       P = 4x

- The constraint for our problem in terms of combined length:

                      L + 4x ≤ 120  

                      L = 120 - 4x Eq1

- The volume - "V" -of the rectangular package with a square cross section is given as:

                      V = L*x^2   ... Eq2

- Substitute Eq1 into Eq2 and form a single variable function of volume "V":

                      V(x) = 120*x^2 - 4x^3

- We are asked to maximize the Volume - " V(x) " - i.e we are to evaluate the critical value of "x" by setting the first derivative of the Volume function to zero:

                     d [ V(x) ] / dx = 240x - 12x^2

                      240x - 12x^2 = 0

                      x*(240 - 12x) = 0

                      x = 0,  x = 20 in

- We will plug in each critical value of "x" back in function " V(x) ":

                      V (0) = 0

                      V(20) = 120(20)^2 - 4(20)^3

                                = 16,000 in^3

- The maximizing dimension of cross section is x = 20 in, the length of the parcel can be determined by the given constraint Eq1:

                       L = 120 - 4*20

                       L = 40 in

- The maximum volume of the rectangular package is with Length L = 40 in and cross section of Ax = ( 20 x 20 ):

the first one, all those points on the line just make a strait line.