# 17. Volume of the Japanese rice box π―π΅

This is a box that the Japanese call “masu” and it is also made by folding paper, i.e. the origami technique. It is easily folded in just a few steps starting with a square shaped paper page that we can label with a. These boxes are traditionally used to measure and store rice. And if we pack rice, it is important for us to fit as much rice as possible in the box. Let’s then determine the height of the box for which the volume is maximum. To get an idea, let’s look at the box in the virtual world with the help of Geogebra. Look, this box of ours is actually a geometric body, which corresponds to a prism. It is known that the prism width is calculated as the product of the surface of the base and the height of the prism (V=BH, where B is the surface of the base of the prism and H is the height of the prism). In that case, the volume of our box can be described as V(H) =a^{2}H-4aH^{2}+4H^{3}, because its base is a-2H, and its height is H. To determine the height for which it has the largest volume, we need to remember calculus. We know that the maximum of a function is determined with the help of the first derivative. Having that in mind, the height of the box for which the volume is maximum is one-sixth of the side of the paper square from which it is composed. Now that we know for which height the volume of the box is the largest, and we need that important when we pack the rice from the beginning of the story that is slowly steamed and prepared for serving.

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