Stage 1 Mathematics method PDF

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Stage 1 Mathematics

Introduction to Differential Calculus – Cake Tin

Yolanda Sitepu

Introduction In this world, there are many practical scenarios where we need to find the maximum or minimum value of a function. This solution is often referred to as the optimum solution, and the process is called optimisation. A cake tin manufacturer will be making cake tins ranging in size from “tiny” to “gigantic”. Some tins will be square based, others will be rectangular based. In all cases the manufacturer wants to maximise the volume of each cake tin. Aim The aim of this task is to investigate the size of a square cut into a piece of tinplate to form an open top cake tin which optimises the volume using various mathematical calculations involving differential calculus, quadratic formula, and developed a conjecture that will be approved with an accurate result. Method In this task there are two parts of investigation. Part A is mainly focuses on determining the size of the squared that based on the cake tin, while in Part B, this part is focusing on the rectangular tins. Part A: Investigating a square tinplate An open top cake tin is to be made by cutting a square from each corner of a square piece of tinplate with side lengths l cm. Once the cut is made the sides are folded to form an open top cake tin. Let x cm be the side length of the square cuts to be made.

l cm

x c l cm

Part A is split into three other sections numbered.

L=5−2 x The limitation of this part is

W =L=5−2 x

H=x L>0 cm since

L∧x arelengths ;

x 0 cm, x > 0 cm since r, s and x are lengths; x < 1 cm if s = 2-2xcm ∴ 0 0 cm, x > 0 cm since r, s and x are lengths; x < 3 cm if s = 6-2x cm ∴0...