Question

calculus, integrals area and volume
show all your solution clearly, please help, thx very much

Task College requires a bottle, which will hold 750mL of liquid, for its official functions. This must be a runctional bottle, suitable for storing and serving the contents. Your task is to design such a bottle. You will need to construct a graph consisting of at least one, but no more than four curved functions (one of which should be degree 3, or higher), together with linear or other functions, that can be rotated around the x-axis (or y-axis if you prefer) so that the volume of the solid of revolution formed gives a bottle suitable for holding 750 mL of liquid. You may also like to use functions studied in this course such as exponential and trigonometric etc. Guidelines This assignment may be completed in pairs-hand in one solution per pair . You must show all working. Include your functions and the domain for each, points of intersection of the functions, calculus and numerical processes used You should use graphing programs and CAS to assist with the graph and calculations. Show the equations to be evaluated but you may use CAS for the graphing calculations. Include a 3D sketch of your bottle. . Your mark will depend on your calculus skills that you are able to demonstrate as well as the accuracy of your results. You must be able to handle the calculus of all chosen functions-do not use functions that have not been covered in the course. i.e. show your working with the calculus! Criteria The marking rubric is on the reverse side of this sheet. It provides you with valuable information on the requirements for this task. Marks Mathematical skills and Total Presentation and communication concepts /60 30 /30 Marking Rubric Presentation and communication 1. Each of the following items need to be presented clearly with suitable headings. /5 Graph of composite functions /5 3D sketch of bottle Calculus processes /5 Numerical work Total /25 2. Conciseness and clarity of report The overall assignment must be well organised, easy to follow, with suitable (brief) supporting sentences presented in a concise, coherent form. That is, order your functions as per the graph. Your calculations should follow the same sequence as the graph. /5 Mathematical Skills and Content 1Shape of bottle appropriate and creative /5 Graph of composite function drawn accurately, showing /6 intersection points Functions allow for breadth of calculus skills to be /6 demonstrated 4 Calculus processes /6 Numerical calculations and Capacity of volume suitable for holding 750 mL of liquid with small air space /5 6 Correct units for volume and capacity Total Hints: Your bottle should have the follow properties (all demonstrated): made of at least three different functions over stated domains two non-linear functions o o one function non-polynomic o continuous your functions must be equal at the edges of the domains no sharp corners-the first derivative of your functions must be equal at the edges of the domains your bottle must be parallel to the axis of rotation at the two furthest ends the volume of your bottle (the volume of revolution) must be 75OmL. (Values between 749.5 and 750.5mL will be suitable) your bottle must be bottle-shaped. i.e. it should have a thin neck and have a reasonable sized base to stand on. Artistic wiggles are fine and are expected a list of your functions and their domains a neat graph of your combined function with points at the edge of domains clearly labelled calculations showing that your wine bottle is: o continuous o free of sharp corners at each domain edge o parallel to the axis of rotation at its two ends calculations determining the volume of your wine bottle Demonstrating a greater expertise and understanding of integral and differential calculus will achieve higher results. Some possible ways could include: use of more than 3 functions a use of difficult or unusual non-polynomic function(s non-stationary edges of domains exact values shown in calculations Calculus calculations shown fully worked

Answer 1

V olurme ef borfe 尺 So, vdume sf beetHe uin be 277 ナ 0.4s lihe 3

edi o 2 2 husturm So, 3 rustur) 2 竹ん SO/ ナグ々2