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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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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Answer 1 · 2015-02-27T15:49:02

First you set up your variables and formulae:

Let $r =$ $r=$ radius and $l =$ $l=$ length

Then girth $G = 2 π r$ $G=2pir$
Volume $V = π r^{2} l$ $V=pir^2l$

We know that $G + l = 2 π r + l \leq 108$ $G+l=2pir+l<=108$ as per regulation.
We will take $= 108$ $=108$ to get the maximum and 'lose' the $l$ $l$
$l = 108 - 2 π r$ $l=108-2pir$

We can now express the volume in $r$ $r$ only:
$V = π r^{2} \cdot l = π r^{2} \cdot (108 - 2 π r) = 108 π r^{2} - 2 π^{2} r^{3}$ $V=pir^2*l=pir^2*(108-2pir)=108pir^2-2pi^2r^3$

To maximise we set the derivative to $0$ $0$

$V'=216pir-6pi^2r^2=6pir*(36-pir)=0 ->$
$36-pir=0->r=36//pi~~11.46$

Substituting:
$G=2pir=72$
$l=108-72=36$
$V=pi*11.46^2*36=14853$

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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?

1 Answer

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