According to postal regulations, 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?
I got:
2r+1=108→Length=108-2r
V(d)= πr^2(108-2r)=108πr^2-2πr^3
V’(d)=216 πr-6 πr^2=0
r(36-r)=0
So , r=36, Length = 36
Is this Right?
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.