Question 1195968
A parcel delivery service accepts cylindrical packages whose length L, plus girth, does not exceed 120 inches.
The girth will be the circumference of the cylinder, therefore
{{{(2*pi*r) + L = 120}}}
or
{{{L = 120 - (2*pi*r)}}}
 A shipper who uses cylindrical cartons, perforated, wishes to design to a carton with maximum ventilation (area).
 What should be the length and radius of the carton?
We can ignore the area of the ends
{{{SA = (2*pi*r)*L}}} 
replace L
{{{SA = (2*pi*r)*(120-2*pi*r)}}}
{{{SA = (240*pi*r) - (4*pi^2*r^2)}}}
arrange like a quadratic equation
{{{-(4*pi^2*r^2) + 240*pi*r}}} = 0
using the axis of symmetry (max area occurs) x = b/(2a), where
{{{a = -4*pi^2}}}
{{{b = 240*pi}}}
{{{r = (-240*pi^2)/(2*-4*pi)}}}
{{{r = (-240*pi^2)/(-8*pi)}}}
Cancel -8 and pi
{{{r = 30/pi}}}
r = 9.55 inches is the radius for max surface area
:
find the length
{{{L = 120 - (2pi*9.55)}}}
L = 60 inches is the length