1. 10 pts postal regulations require that for priority mail the length and girth of a rectangular parcel…

Explanation:

Step1: Define variables

Let the side length of the square base be (x) inches and the length of the parcel be (y) inches. The girth (G = 4x), and the constraint is (4x + y=108), so (y = 108 - 4x). The volume (V) of the rectangular - parcel (with square base) is (V=x\times x\times y=x^{2}y). Substitute (y = 108 - 4x) into the volume formula: (V(x)=x^{2}(108 - 4x)=108x^{2}-4x^{3}), where (x>0) and (y = 108 - 4x>0) (so (x < 27)).

Step2: Find the derivative of the volume function

Use the power rule ((x^{n})^\prime=nx^{n - 1}). (V^\prime(x)=\frac{d}{dx}(108x^{2}-4x^{3})). (V^\prime(x)=216x-12x^{2}=12x(18 - x)).

Step3: Find the critical points

Set (V^\prime(x)=0). (12x(18 - x)=0). Using the zero - product property (a\times b = 0) implies (a = 0) or (b = 0). (x = 0) or (x = 18). Since (x>0) and (x<27), we consider (x = 18).

Step4: Use the second - derivative test

Find the second - derivative (V^{\prime\prime}(x)=\frac{d}{dx}(216x - 12x^{2})). (V^{\prime\prime}(x)=216-24x). When (x = 18), (V^{\prime\prime}(18)=216-24\times18=216 - 432=-216<0). So the function (V(x)) has a local maximum at (x = 18).

Step5: Calculate the maximum volume

When (x = 18), find (y) using (y = 108 - 4x). (y=108-4\times18=108 - 72 = 36). The volume (V=x^{2}y). Substitute (x = 18) and (y = 36) into the volume formula: (V=(18)^{2}\times36=324\times36 = 11664) cubic inches.

Answer:

The maximum possible volume of the rectangular parcel is (11664) cubic inches.