Therefore, the required dimensions for the cardboard areįor this exercise, we use hypothetical dimensions for the Pizza Box problem. We then take the value of V'(1) = 0 whenever the value of I = m. ![]() Suppose cardboard of A square inches is to be used for the construction of a Pizza Box by means of the Pizza Box methodology and fixing the height of the Pizza Box at T inches, we must find the dimensions of the rectangle that if used would optimize or maximize the volume of the Pizza Box ( Optimization Problem, 2011).įrom figure 1, we can take w to be equal to A/l. The approach that was considered for this process was as elaborated in the next paragraph. In order to allow for variable rectangular pieces’ dimensions as well as various sizes of the corner pieces, calculus must be used and it must involve various variables. However, this is not applicable in a real word situation. With a little iteration, researchers have found that restricting the rectangular piece of cardboard’s shape effectively limits the box’s maximum volume (Daley et al., 2015). In this problem, we consider the value of T, which is the volume of the box, to be half as large by means of the Pizza Box technique, and the maximum is considered to occur at the same value of T in all of the cases. The ensuing volume is given by the following formula:įor a value of T < w/2, for the Pizza Box construction, its volume is computed as follows In order to construct the pizza box, we adopted a no-top construction methodology by cutting out T by T-long squares from the cardboard’s rectangle having a length I as well as width w, where w < I. If we cut the cardboard along the solid lines and then appropriately fold it along the dotted lines then the use of appropriately-placed staples can secure the box to be fairly usable. The most suitable method of construction is shown in figure 1 on the next page. As noted by Dundas, most calculus students have been faced with the problem of finding the optimum volume of a box that is constructed from rectangular cardboard pieces by cutting equal squares from every corner, and then, folding up the cardboard’s sides (1984). ![]() In 3D packaging problems, some studies assume the existence of an unlimited capacity (Electric Teaching, 2013). Normally, optimization as well as heuristic approaches have been applied in solving Pizza Box optimization problems/issues. Notably, the overall effect is a reduction in the production cost of the container or pizza box, as well as a reduction in the cost of transportation or shipment. The main problem of fitting the products is often referred to as a 3-dimensional packing problem (Eley, 2002 Pisinger, 2002). Manufacturers and Pizza firms have to ensure the products fit in the 3D box vessels having optimal placement in order to ship or transport products at the lowest cost possible. According to Delfour and Zolésio (2001), getting a higher volume or occupancy rate of a given container is a vital goal for most businesses dealing in packaging and shipping of goods. Calculus can be used as a tool to maximize, or in some cases minimize (technically referred to as optimize) a situation. In this case, the use of minimum cardboard material will be used to manufacture a pizza box having the largest volume. According to Daley, Gutierrez, and Wilkerson (2015), optimization is a concept that requires the use of minimal materials for maximum effect. One of the grand uses of calculus in the real-world setting is optimization. In this report, we describe the application of Calculus in solving of the pizza box optimization problem. Undeniably, calculus can be useful in devising efficient as well as practical construction approaches. ![]() This kind of problem can be solved through an expert application of Calculus. This problem involves sticking equal squares from each corner of the rectangle and then folding up the resulting sides as described by Dundas (1984). Most companies and organizations have been faced with the problem of finding the maximum volume of a box that is constructed from rectangular pieces of cardboard.
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