SOLUTION: A parcel delivery service accepts cylindrical packages whose length L, plus girth, does not exceed 120 inches. A shipper who uses cylindrical cartons, perforated, wishes to design

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Question 1195968: A parcel delivery service accepts cylindrical packages whose length L, plus girth, does not exceed 120 inches. 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?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
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
%282%2Api%2Ar%29+%2B+L+=+120
or
L+=+120+-+%282%2Api%2Ar%29
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+=+%282%2Api%2Ar%29%2AL
replace L
SA+=+%282%2Api%2Ar%29%2A%28120-2%2Api%2Ar%29
SA+=+%28240%2Api%2Ar%29+-+%284%2Api%5E2%2Ar%5E2%29
arrange like a quadratic equation
-%284%2Api%5E2%2Ar%5E2%29+%2B+240%2Api%2Ar = 0
using the axis of symmetry (max area occurs) x = b/(2a), where
a+=+-4%2Api%5E2
b+=+240%2Api
r+=+%28-240%2Api%5E2%29%2F%282%2A-4%2Api%29
r+=+%28-240%2Api%5E2%29%2F%28-8%2Api%29
Cancel -8 and pi
r+=+30%2Fpi
r = 9.55 inches is the radius for max surface area
:
find the length
L+=+120+-+%282pi%2A9.55%29
L = 60 inches is the length