Estimating The Volume of a 3D Item That has Symmetry of Rotation - Mathematics Assignment Help

Assignment Task:

Task:

Purpose
The purpose of this task is to investigate relationships between the dimensions of an object and its mathematical representation.
Students will gain a conceptual grasp of calculus, and ability to use its techniques in applications.
Assessment Type: Mathematical Investigation Task Weighting 20%

Task Summary

• When a function y=f(x) for a?x?bis rotated about the x-axis by 360? it sweeps out a solid shape called a solid of revolution.
• Consider the graph of the function y=x+3 for -3?x?3 as shown in Figure 1. When the graph of the function is rotated about the x-axis by 3600 it will produce a solid cone with a height, h, of 6 units and a radius, r, of 3 units. The volume of the solid cone can be determined by the formula V=13h?r2.
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• 258486339842Figure 1
• 0Figure 1
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• Now, consider the graph of y=-x2+9 for 0?x?3 as shown in Figure 2. When the graph of the function is rotated about the x-axis by 3600 it will produce a solid shape with a height, h, of 3 units and a radius, r, of 9 units. The volume of the solid can only be determined by calculus techniques.
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• 2322151118450Figure 2
• 0Figure 2
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• Using calculus, the volume (V) of such solid can be calculated by V=?ab[f(x)]2dxPart 1: Measuring the overall volume of a given 3D item
• The shape represented on the axes below is made up of two separate functions, f(x)=1+1-x and g(x)=1-1-x.
• Show that the two functions intersect at (1, 1).
• Complete the drawing of the lower section of the solid of revolution by rotating the two functions about the x-axis by 360?.The integration statement that represents the volume of this particular solid of revolution is V=?01[f(x)]2dx-?01[g(x)]2dxShow that the formula for the volume simplifies to V=4?011-x dx.
• Hence, algebraically calculate the exact value of the solid of revolution.
• Part 2: Measuring the overall volume of a 3D item (that has symmetry of rotation)
• Find an item at home that has symmetry of rotation. Measure the outside dimensions of this item as accurately as you can. If your item is hollow, you may consider measuring its inside dimensions as well. Record carefully how you made these measurements and discuss any limitations to your method.
• Use displacement to measure the volume of your item. Describe carefully and concisely how you did this.
• Part 3: Estimating the volume of a 3D item (that has symmetry of rotation), using Integration
• Represent your 3D item as a series of 2D functions to best replicate the outside (and inside if hollow) dimensions of the item of your choice. The rotation of these series of functions should closely represent this item.
• Use integral calculus and the volume of rotation practiced in Part 1of this investigation to calculate the volume of your item. Describe carefully and clearly how you do this.
• Discuss in detail any limitations to the development and effective application of your mathematical model.
• The focus of this investigation is to discuss, demonstrate and develop the mathematics behind your model. You will need to consider and demonstrate your process when deciding things such as:
• what functions to use to model the shape of your item
• the method you applied to connect your functions
• the accuracy of your integral calculations
• You are expected to demonstrate the use of technology in this project.
• You are strongly advised to read the attached performance standards that this task will be assessed against before you begin this task.
• Assessment Guide
• In constructing your investigation, you need to use the following structure:
• Introduction
• Define the problem that has been presented to you (the ‘what’)
• Describe clearly and concisely how this task is applicable to your learning and to the real world (the ‘why’- the significance of the task)
• List all methods you are going to demonstrate to solve this problem, including any use of technology (the ‘how’)
• Mathematical Calculations & Analysis
• Follow all the instructions provided with your problem. Provide answers to all questions. Use headings where appropriate.
• Provide relevant diagrams or graphs that could help illustrate any concepts you are explaining
• Clearly explain each step in your mathematical reasoning with notes or short sentences. Justify all choices you make when answering questions
• Analyse and compare the two methods used to determine the volume of your item. Discuss any differences in the outcomes.
• Discuss the reasonableness and limitations of the mathematical results in your investigation

Conclusion
Summarise your findings by:

• Outlining any observations or findings you have made
• Clearly describe and illustrate your final results
• Suggest any alterations you could make to improve the accuracy of your model.
• Appendices
• Includes any additional information you would like to include that strengthens your justification or documents process that is not directly essential in your analysis section.
• You are expected to provide a bibliography for any resources you have used outside of the Mathematical Methods course during this task.
• Submission Requirements Modified
• Format (including text type) Written report ?Word Count / Length Maximum 15 A4 pages
• Minimum font size 10 Deadlines Date Submission Requirements
• Checkpoint #1 Checkpoint #2 Due Assessment Design CriteriaWhat you will be assessed on Task Specific Clarification To meet the level descriptors make sure you:
• Concepts and Techniques
• Knowledge and understanding of concepts and relationships.
• Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts.
• Application of mathematical models.
• Use of electronic technology to find solutions to mathematical problems.
• Complete, concise, and accurate solutions to determine the volume of the item of your choice using both measurement and integral calculus methods.
• Appropriate selection of techniques to measure the volume of a chosen 3D item with rotational symmetry.
• Development and effective application of functions to model the item of choice once rotated.
• Efficient use of technology (EFOFEX and graphics calculators) to calculate the volume of a chosen 3D item with rotational symmetry.
• Reasoning and Communication
• Interpretation of mathematical results.
• Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations.
• Use of appropriate mathematical notation, representations, and terminology.
• Communication of mathematical ideas and reasoning to develop logical arguments.
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• Clear, efficient and precise interpretation of the mathematical results. All choices of functions used clearly justified.
• In-depth discussion of any limitations to the model developed or techniques used to determine the volume of your item and the reasonableness of the results.
• Report constructed in the appropriate format. All graphs clearly labelled. Correct notation proficiently used for all calculations.
• Mathematical arguments clear and easy to follow.
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• Concepts and TechniquesReasoning and CommunicationAComprehensive knowledge and understanding of concepts and relationships.
• Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts.
• Successful development and application of mathematical models to find concise and accurate solutions.
• Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. Comprehensive interpretation of mathematical results in the context of the problem.
• Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations.
• Proficient and accurate use of appropriate mathematical notation, representations, and terminology.
• Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments.
• Effective development and testing of valid conjectures, with proof.
• BSome depth of knowledge and understanding of concepts and relationships.
• Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts.
• Some development and successful application of mathematical models to find mostly accurate solutions.
• Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. Mostly appropriate interpretation of mathematical results in the context of the problem.
• Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations.
• Mostly accurate use of appropriate mathematical notation, representations, and terminology.
• Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments.
• Mostly effective development and testing of valid conjectures, with substantial attempt at proof.
• CGenerally competent knowledge and understanding of concepts and relationships.
• Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts.
• Successful application of mathematical models to find generally accurate solutions.
• Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. Generally appropriate interpretation of mathematical results in the context of the problem.
• Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations.
• Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy.
• Generally effective communication of mathematical ideas and reasoning to develop some logical arguments.
• Development and testing of generally valid conjectures, with some attempt at proof.
• DBasic knowledge and some understanding of concepts and relationships.
• Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts.
• Some application of mathematical models to find some accurate or partially accurate solutions.
• Some appropriate use of electronic technology to find some accurate solutions to routine problems. Some interpretation of mathematical results.
• Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations.
• Some appropriate use of mathematical notation, representations, and terminology, with some accuracy.
• Some communication of mathematical ideas, with attempted reasoning and/or arguments.
• Attempted development or testing of a reasonable conjecture.
• ELimited knowledge or understanding of concepts and relationships.
• Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems.
• Attempted application of mathematical models, with limited accuracy.
• Attempted use of electronic technology, with limited accuracy in solving routine problems. Limited interpretation of mathematical results.
• Limited understanding of the meaning of mathematical results, and their reasonableness or limitations.
• Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy.
• Attempted communication of mathematical ideas, with limited reasoning.
• Limited attempt to develop or test a conjecture.

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