Question

Suppose that postal requirements specify that parcels must have length plus girth at most 78 inches. Consider the problem of finding the dimensions of the square ended rectangular package of greatest volume that is mailable. A rectangular prism has length labeled X, width labeled h, and height labeled x.

Required:
a. Express the length plus the girth in terms of x and h.
b. Determine the objective and constraint equations.
c. What is the constraint equation?

Answer 1

A. 4x + h

B. 4x + h = 78

V = x²h

Explanation:

1. The length plus girth

X +x+x+x +h

= 4x +h

2. Constraint equation

4x + h = 78 ----(1)

Objective function

Volume = v = x*x*h

V = x²h ----(2)

3. Quantity as function of x

From equation 1 in answer part b

4x+h = 78

We make h subject

h = 78-4x

We put values of h in equation 2 in part b

V = x²h

V = x²(78-4x)

V = 78x²-4x³

To get values of x and h

V = 78x²-4x³

Dv/dx = 0

= 156x - 12x²

156x = 12x²

Divide through by 12x

156x/12x = 12x²/12x

x = 13

h = 78-4x

h = 78 - 4(13)

h = 78 - 52

h = 26 inches