What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches?

koraby2bc 2022-09-23 Answered
What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches?
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Answers (1)

Hayden Espinoza
Answered 2022-09-24 Author has 7 answers
First you set up your variables and formulae:
Let r=radius and l=length
Then girth G = 2 π r
Volume V = π r 2 l
We know that G + l = 2 π r + l 108 as per regulation.
We will take =108 to get the maximum and 'lose' the l
l = 108 - 2 π r
We can now express the volume in r only:
V = π r 2 l = π r 2 ( 108 - 2 π r ) = 108 π r 2 - 2 π 2 r 3
To maximise we set the derivative to 0
V = 216 π r - 6 π 2 r 2 = 6 π r ( 36 - π r ) = 0
36 - π r = 0 r = 36 / π 11.46
Substituting:
G = 2 π r = 72
l = 108 - 72 = 36
V = π 11.46 2 36 = 14853
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