Question

suppose that postal requirements specify that parcels must have length plus girth at most 84 inches. consider the problem of finding the dimensions of the square - ended rectangular package of greatest volume that is mailable. each square end is labeled with x and the remaining dimension is labeled with h. (a) express the length plus the girth in terms of x and h. 4x + h (b) determine the objective and constraint equations. what is the constraint equation?

Explanation:

Step1: Identify the constraint

The postal requirement states that length plus girth is at most 84 inches. Since length plus girth is \(4x + h\), the constraint equation is \(4x+h\leq84\). For finding the maximum - volume, we consider the equality case \(4x + h=84\), so \(h = 84 - 4x\).

Step2: Define the objective function

The volume \(V\) of a rectangular box with square base of side - length \(x\) and height \(h\) is \(V=x\times x\times h=x^{2}h\). Substitute \(h = 84 - 4x\) into the volume formula, we get \(V(x)=x^{2}(84 - 4x)=84x^{2}-4x^{3}\).

Answer:

The constraint equation is \(4x + h=84\)