Maximize Luggage Space

I can't find it now, but there was once an airline that measured carryon luggage a really interesting way to provide more flexibility. Basically, you measure the length, width, and height of your luggage and add them up all together. That total length could not exceed a certain length in order to be carried on the flight. So, how could you maximize your luggage space?

Overengineering the Obvious Solution

Let's say the maximum length is $L$. We already know that the sum of the dimensions cannot exceed that length, therefore $l+w+h \le L$ or $h \le L-l-w$. The volume of the luggage is $V=l \times w \times h$ or by substituting we get $V=l \times w \times (L - l - w)$. After simplifying, this is our formula for the volume of your luggage with total dimension length $L$.

$$V = lwL - l^{2}w - lw^{2}$$

We can then find the maximum volume by taking the partial derivatives with respect to $l$ and $w$ and setting them to $0$.

$$\frac{\delta V}{\delta l} = wL - 2lw - w^{2} = 0$$

$$L - 2l - w = 0\text{ (1)}$$

$$\frac{\delta V}{\delta w} = lL - l^{2} - 2lw = 0$$

$$L - l - 2w = 0\text{ (2)}$$

And by putting equations $(1)$ and $(2)$ together:

$$l + 2w = 2l + w$$

$$w = l$$

No surprise there, but how does that relate to $L$ and what on earth is the height $h$? Let's just rearrange $(1)$ for example.

$$L = 2l + w = 3l = 3w$$

That means the length and width are one-third of the total length which is $\frac{L}{3}$. Going all the way back to the beginning, we can now calculate height $h$.

$$h \le L - l - w = L - \frac{L}{3} - \frac{L}{3} = \frac{L}{3}$$

This tells us that the height is also one-third of the total length $\frac{L}{3}$. The volume of your "ideal" luggage is length times width times height, so we can multiply that out in terms of $L$.

$$V = l \times w \times h = \frac{L}{3} \times \frac{L}{3} \times \frac{L}{3} = \frac{L^{3}}{27}$$

Conclusion

If your maximum total dimension is 60cm (for example), then each side length of your luggage should be 20cm to maximize the volume. You could store up to 8,000 $\text{cm}^{3}$. You could pour over two gallons of milk into your suitcase before it fills up! But always check your airline's luggage rules before flying. I doubt they'll let you on board with a 20 by 20 by 20 cm milk cube, but good luck, and happy travels!