A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 105 inches.

Question

A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross-section) of 105 inches.

Find the dimensions of the package of maximum volume that can be sent. (The cross-section is circular.)

radius?

Length?

Answer 1

Pi = 3.14

2(Pi)r + h = 105 in => h = 105 - 2(Pi)r

V = (Pi) r²h => V= (Pi) r² * [ 105 - 2(Pi)r ] = 105 (Pi) r² - 2(Pi)²r³

dV/dr = 210(Pi)r - 6 (Pi)² r² = 0 => 6(Pi)r [35 - (Pi)r] = 0 =>

r=0 which is irrelevant.

Or

[35 - (Pi)r] = 0 => 35 = (Pi)r => r= 35/(Pi) in =>

h=105 -2(Pi) [35/(Pi)] = 105 - 70 = 35 in