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:


Expert Answers

An illustration of the letter 'A' in a speech bubbles

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

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial