Question

Some airlines have restrictions on the size of items of luggage that passengers are allowed to take with them. Suppose that one has a rule that the sum of the length, width and height of any piece of luggage must be less than or equal to 222 cm. A passenger wants to take a box of the maximum allowable volume. If the length and width are to be equal, what should the dimensions be?

Answer 1

The dimensions should be length=width=height=74 cm.

Explanation:

We have an optimization with restriction problem.

We have to maximize the volume, subject to restriction in the sum of the length, width and height.

Let x be the length and width, that are equal, and z be the height.

The restriction can be expressed as:

`x+x+z\leq222\\\\2x+z\leq222`

We can express z in function of x as:

`2x+z=222\\\\z=222-2x`

The volume, the function to be optimized, can be expressed as:

`V=x^2z=x^2(222-2x)=222x^2-2x^3`

To optimize, we derive and equal to zero.

`(dV)/(dx)=(d)/(dx)[222x^2-2x^3]=2*222x-3*2x^2=444x-6x^2=0\\\\\\444x-6x^2=0\\\\x(444-6x)=0\\\\444-6x=0\\\\x=444/6=74`

We have the optimum length. We can now calculate the height z:

`z=222-2(74)=222-148=74`