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Introduction to Differential Calculus – Cake Tin
发布时间：2024-05-27
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STAGE 1 MATHEMATICS
Assessment Type 2: Mathematical Investigation
Introduction to Differential Calculus - Cake Tin
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Task Description: A cake tin manufacturer makes 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. Your task is to investigate the size of a rectangle cut into a piece of tinplate to forman open top cake tin which optimises the volume. Through the use of mathematical calculations, involving calculus, some conjectures will be made. Present your findings as a formal mathematical report using the headings • INTRODUCTION • MATHEMATICAL INVESTIGATIONS & ANALYSIS • CONCLUSION |
Part 1 - 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 forman open top cake tin. Let x cm be the side length of the square cuts to be made.
• Given the length of each side of the tinplate is 5cm, show that the volume of the cake tin can be expressed as
V(x) = 25x − 20x2 + 4x3 cm3
• Hence, use calculus techniques to find the exact value of x that will maximise the volume of the cake tin.
• Repeat the above process for at least two other side lengths
. Make a conjecture based on the size of the squares to be cut from a square piece of tinplate with side lengths lcm that will maximise the volume of the cake tin.
. Prove your conjecture.
. Discuss any limitations and assumptions to your findings.
Part 2 - Rectangular Tinplate
Consider the following rectangular piece of tinplate with sides T cm and s cm. An open top cake tin is to be made by cutting a square (xcm by xcm) from each corner.
. Using a ratio of 1:2 for the sides of the rectangle, investigate the relationship between x (the cut to be made for the square) and the length of each side of the rectangle such that the cake tin has a maximum volume. [hint: consider a few specific cases]
. Summarise your findings and comment on any assumptions made and/or any limitations to your findings.
Notes
Your report on the mathematical investigation should include the following:
• an outline of the problem and context
• the method required to find a solution, in terms of the mathematical model or strategy used
• the application of the mathematical model or strategy, including:
o relevant data and/or information
o mathematical calculations and results, using appropriate representations
o the analysis and interpretation of results, including consideration of the reasonableness and limitations of the results
• the results and conclusions in the context of the problem
• a bibliography and appendices, as appropriate.
The format of an investigation report maybe written or multimodal.
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ASSESSMENT |
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Your report will be graded against the following Stage 1 Mathematics Performance Standards using the rubric attached. Concepts and Techniques The specific features areas follows: CT1 Knowledge and understanding of concepts and relationships. CT2 Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts. CT3 Application of mathematical models. CT4 Use of electronic technology to find solutions to mathematical problems. Reasoning and Communication The specific features areas follows: RC1 Interpretation of mathematical results. RC2 Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations. RC3 Use of appropriate mathematical notation, representations, and terminology. RC4 Communication of mathematical ideas and reasoning to develop logical arguments. RC5 Development and testing of valid conjectures. |
