The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 60 inches. Find the dimensions of the rectangular package of largest volume that may be sent.
A)x = 7.5, y = 7.5, z = 20
B)x = 5, y = 5, z = 40
C)x = 10 , y = 10 , z = 20
D)x = 4, y = 6, z = 40
E)x = 2, y = 12, z = 32

Question

Quizplus · Accepted Answer

Let the length and width of the rectangular package be x and y, respectively. Then we have the equation:
2x + 2y + z = 60
Rearranging, we have:
z = 60 - 2x - 2y
The volume of the package is given by:
V = xyz
Substituting for z, we have:
V = xy(60 - 2x - 2y)
Expanding and simplifying, we have:
V = -2x^3 - 2y^3 + 60xy^2 + 60x^2y
To find the maximum volume, we take partial derivatives with respect to x and y and set them equal to zero:
∂V/∂x = -6x^2 + 120y - 120xy = 0
∂V/∂y = -6y^2 + 120x - 120xy = 0
Solving these equations simultaneously, we get:
x = y = 10
Substituting into the expression for z, we get z = 20. Therefore, the dimensions of the rectangular package of largest volume that may be sent are x = 10, y = 10, and z = 20. So, the best choice is C.

The Sum of the Length (Denote by Z) and the Girth