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)
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