The girth plus length of parcels sent by fourth-class mail may not exceed 108 inches What is the largest volume of a cylindrical parcel that can be sent by fourth-class mail?

Answer 1

girth = 2PI r

height=h

h+2 PI r =108

h=108-2 PI r

Volume = PI r^2 h

V = PI r^2 (108-2PI r)

Differentiate with respect to r

dV/dr = PI (2r) (108-2PI r) +PI r^2 (-2PI) =0

216 PI r -4 PI^2 r^2 -2 PI^2 r^2=0

216 PI r -6 PI ^2 r^2=0

PI r [ 216-6 PI r] =0

6 PI r = 216

r = 36/PI --- dimensions

h = (108-2 PI (36/PI)) = 108 -72=36 -- dimensions

Largest volume = PI r^2 h = PI (36/PI)^2 (36)

=1,296 / PI

d^2V/dr^2 =216 PI -6 PI^2 (2r)

when r=36/PI, d^2V/dr2 = 216 PI - 6PI^2 (2)(36)/PI

=216 PI -432 PI < 0, indicates V has been maximized.