What values for the dimensions of the package lead to maximum volume given a constraint on length + girth?

Suppose postal requirements are that the maximum of the length plus the girth (cross sectional perimeter) of a rectangular package that may be sent is 275 inches. Find the dimensions of the package with square ends whose volume is to be maximum.

(leave your answers to 3 decimal places)

Square side:

Length:

Expert Answers

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If `a` is the length of one of the square sides and `l` is the length of the package, we have that the required girth `G` satisfies

` ``G = l +4a`

We want the girth to be as big as possible as this will lead to a bigger volume, so we have that `G=275` (the maximum girth allowed).

The volume `V` of the package satisfies

`V = la^2`

To maximise `V` with respect to `G = 275` we use the method of Lagrange multipliers

This involves solving the simultaneous equations

1) `(del V)/(del l) - lambda(delG)/(del l) = 0`

2) `(delV)/(dela) -lambda(delG)/(dela) = 0`

for `lambda`, subject to the constraint `G=275`

Now we have

1) `a^2 - lambda(1) = 0`

2) `2al - 4lambda = 0`

 

` ` 1) `implies`  `a^2 = lambda`  `implies` `a=sqrt(lambda)`

Substituting into 2) we get

2)  `2lsqrt(lambda) -4lambda = 0` `implies` `l = (4lambda)/(2sqrt(lambda)) = 2sqrt(lambda)`

Considering the constraint `G = l +4a = 275`  we have that

`2sqrt(lambda)+4sqrt(lambda) = 275`

`implies`  `sqrt(lambda) = 275/6`

Therefore `a=275/6`  and `l = 275/3` to give maximum volume `V=la^2`

a = 45.833, b = 91.667

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