CALCULUS AND QUADRATIC FORMULA TO DETERMINE LENGTHS

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Calculus, which is one of the foremost branches of Mathematics, studies about the rates of changes. It was invented by Isaac Newton and Gottfried Leibniz in the latter half of the 17th century. It is now used in different fields and subjects such as science, economics, and engineering (Wikipedia, 2019).  For example, in Physics, calculus is used in many concepts, including motion, electricity, heat, and light. In the field of Chemistry, calculus can be used to predict reaction rates and radioactive decay. Moreover, in Biology, it is utilised to form birth and death rates. (teAchnology,2019). Calculus can also solve the problems of optimising something. Optimisation is one of the common uses of calculus in practical situations.

In real life, cakes come in many different shapes and sizes. Therefore, different cakes need to be cooked in different cake tins. A cake tin manufacturer, which makes square-based or rectangular based cake tins ranging in size from “tiny” to “gigantic”, wants to optimise the volume of the cake tins. The demand for optimising the volume of the cake tins is that the bigger the volume is, the less amount of material is used so that the manufacturer can get economic benefits. To form an open top cake tin, a square from each corner of a piece of tinplate is cut, then the sides of the tinplate are folded. In this Directed Investigation, the side length of these four squares will be investigated, which value(s) of the side length would maximise the volume of the cake tins. Using various mathematical methods such as calculus and the quadratic formula, some conjectures will be made and proved. This investigation will also discuss the limitations and assumptions.

MATHEMATICAL INVESTIGATIONS & ANALYSIS