Suppose an airline policy states that all the baggage must be boxed shaped with a sum of length, width, and height not exceeding 138 inches. What are the dimensions and volume of a square based box with the greatest volume under these conditions.

Answer 1

let the dimensions by x by x by y

we know x+x+y ≤ 138
2x + y ≤ 138
y ≤ 138-2x

V = x^2 y = x^2(138-2x)
= 138x^2 - 2x^3
dV/dx = 276x - 6x^2
= 0 for a max of V
6x^2 = 276x
6x = 276
x = 46
then y = 138-92 = 46

so the box must be a cube of 46 by 46 by 46
as suspected

notice 46+46+46 ≤ 138