if a package has a combined length and girth of 130 inches. what is the maximum volume

Answer 1

maximum volume is a cube, for a rectangular package

L+W = 130 inches

L=W for a cube

2L = 130

L =130/2 = 65 inches = W = height=length=width or "girth"

65 x 65 x 65 = 4225x 65= 274,625 cubic inches = maximum volume possible

but "girth" is not a math or physics term. It usually refers to a person's waist line, a human's circumference around the waist area.

Other possibilities

Girth could = the circumference of a cylindical package

G= piD where D = the diameter of the package or twice the radius. Length then would = the height of the cylinder G+Height = G+L = D+L = 130 inches

D= 130-L

Volume of the Cylinder = L(D/2)^2(pi)

V= L(130-L)^2(pi)

V= piL(16900-260L + L^2)

V= pi(16900L-260L^2 + L^3)

V' = pi(16900-520L +3L^2) = 0

3L^2 -150L +16,900=0

use the quadratic formula, solve for L

substitute back into the 3rd degree formula for V to find maximum Volume

or a spherical package gives maximum volume

or in packaging terms, for rectangular packages: G=2H+2W = 2L+2W

G+L =130

2L+2W = 130

L+W = 65

L=65-W

Volume = V=(65-W)W^2 = 65W^2-W^3

V' = 130W -3W^2 = 0

W(130-3W)=0

3W = 130

W = 130/3 = 43 1/3 inches

G=65-43 1/3 = 11 2/3

max V= (35/3)(130/3)^2 = 35(16900)/27

= 21,907.4041 in^3

those calculations based on the base of the package having a square shape