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 36 inches.Find the dimensions of the rectangular package of largest volume that may be sent.
B)
C)
D)
E)

Question

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 36 inches.Find the dimensions of the rectangular package of largest volume that may be sent. ​
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

Let the length, width, and height of the package be denoted by x, y, and z, respectively. Then, we know that:
z + 2x + 2y = 36, or equivalently, z = 36 - 2x - 2y
The volume of the package is given by V = xyz.
Substituting z in terms of x and y, we have:
V = xy(36 - 2x - 2y)
Expanding, we get:
V = -2x^2 y - 2xy^2 + 36xy
Taking the partial derivatives with respect to x and y and setting them to zero, we get:
dV/dx = -4xy + 36y = 0
dV/dy = -4yx + 36x = 0
Solving these equations, we get:
x = y = 9
Substituting these values into z = 36 - 2x - 2y, we get:
z = 9
Therefore, the dimensions of the package of largest volume that may be sent are: x = y = 9, z = 9.

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