Final answer:
To determine the volume of a package with USPS restrictions, one must express the length in terms of the width and height, obeying the maximum combined length and girth limit of 108 inches. The domain for the width and height is positive, with width being less than 54 inches and height being less than the difference between 54 and the width. The volume is maximized when dimensions are as equal as possible, respecting the girth limit, hinting towards a more cubic dimension within the restrictions.
Step-by-step explanation:
The U.S. Postal Service has regulations for the maximum combined length and girth of a rectangular package; it must not exceed 108 inches to be sent. To address this in mathematical terms:
(a) Writing the volume as a function:
Let's denote the length of the package as L, width as W, and height as H. The girth is the perimeter of the cross-section perpendicular to the length, which is calculated as 2W + 2H. According to the USPS rules, the sum of the length and girth (L + 2W + 2H) cannot exceed 108 inches. To write the volume V as a function, we assume L as the subject and express it in terms of W and H:
L = 108 - 2W - 2H
The volume V of the package can be written as:
V = L × W × H
V = (108 - 2W - 2H) × W × H
(b) The domain of the function:
The domain of the volume function V(W, H) refers to all possible values that W and H can take. Since the length, width, and height must be positive and within the USPS restrictions:
0 < W, H
0 < L = 108 - 2W - 2H
The domain will be:
0 < W < 54
0 < H < (54 - W)
(c) Graphing the function:
Using a graphing utility to visualize the volume function involves plotting V against values of W and H within their respective domain. The appropriate viewing window should consider the domain restrictions for W and H up to a maximum of 54 inches for W and decreasing from there as a function of W for H.
(d) Maximizing the volume:
The dimensions that maximize the volume of the package can be found by using optimization techniques such as setting the partial derivatives of the volume function with respect to W and H to zero. However, this can be complex without additional tools. Generally, a balanced approach where length is maximized (subject to the perimeter constraints) tends to yield larger volumes.
Without the aid of calculus, we can argue that the package's volume will be maximized when the dimensions are as close to a cube as the restrictions allow, which would be the case when the length is equal to the width plus height (L = W + H) and all dimensions are as large as possible within the girth constraint.