A rectangular piece of card measures $16\text{cm}$ by $10\text{cm}$.

The four shaded squares of side length $x\text{cm}$ are to be cut away, and the card will be folded along the dashed lines to form a cuboid-shaped tray.

Find the largest possible volume of the tray, and the value of $x$ for which this occurs.

Justify, using the second derivative test, that the volume you have found is a maximum.

After folding the card, the tray looks like this:

Give your answer below