Question

International Air Transport Association (IATA) requires that any checked baggage on any flight must

satisfy the following condition: The sum height + width + length is at most 158 cm. Priya wants to

make a rectangular box that she will be able to check in on her next flight. She wants the length of

the box to be twice its width, and she wants to maximize the volume of the box. Determine the width,

length and height of the optimal box. Explain why the volume is maximized. Express the maximum

volume in m .

Answer 1

Frame the problem as:

Max x*y*z

sub. to x+y+z<=158

x=2y

where x denotes length , y denotes width , z denotes height.

The problem is same as

Min -x*y*z -----> f

sub. to x+y+z<=158 ------> g

x=2y -------> h

Since the objective function is non linear , so we can't use the methods studied in linear programming problems.

We'll use Karush-kuhn-tucker in which we used to find K-K-T points (extremas in 2-D) and among those k-k-T's find value of f = -x*y*z , whichever minimum pick them up.

The point you got to minimize - x*y*z will maximize x*y*z and your aim will be in your hands.

(this K-K-T method involves just calculations,you can google the method if not heard before.)