# Midterm Assignment (MA1608)

General Information

This assignment consists of 4 problems and the mark you obtain will contribute 50% to your final mark for this module. You need to attempt all problems for full marks. In your answers, you may use any material that we have covered in lectures, but you should always give your own account. The total number of marks available is 100, including 20 marks allocated specifically for the quality of the text with explanations, typesetting and general presentation. There are no restrictions to the length of your report, but try to be concise and avoid including unnecessary details. When in doubt, please ask. You need to reference the non-trivial information you use (which book, website or class notes, who did you ask etc.). What is important is to be candid about where your information comes from, not to let the reader believe you did something you did not. Note that the values of the various parameters in the assignment are determined individually for you on the basis of your student number . In the assignment, the last three digits of your student number are called S,UandW.

For example, if your student number is 2187650, then the last three digits are 650, and in this case S= 6,U= 5 and W= 0. You must use the numbers corresponding to your student ID. The assignment must be typewritten, using an equation editor for the mathematics . Although the report must be submitted online, it should be prepared in a form suitable for printing. In particular, it must be optimised for A4 sized paper; the minimum font size is 12pt.

You can create your report in Microsoft Word, but it has to be submitted as a single PDF document. Your PDF report has to be submitted as the main document. If you need to submit any other files, they have to be uploaded as a WISEflow attachment in the form of a SINGLE zipped file. For example, if you do some calculations with Matlab, or want to submit some appendices with a scan of your calculations, then the corresponding files have to be uploaded as a single zip file. Please note that the University enforces a very strict Policy for Coursework Submission, as detailed here: Please remember that this is an individual assignment. Group discussions are healthy and encouraged; however, the work you submit must be your own and MUST NOT BE prepared in collaboration with others. Misconduct in assessment is taken very seriously by the University. You are expected to abide by Senate Regulation 6 - Student Conduct (Academic and Non-Academic), which can be found here:

Advice on understanding what plagiarism and collusion are and how they can be avoided can be found here:

Assignment Tasks NOTE: Justify all your answers, provide details of the calculations and explain what is being done.

## Problem 1 20 marks (calculation) + 5 marks (presentation)

Letx, yandzbe positive real numbers. Apply the AM-GM inequality to solve the following tasks. (a) Given that xyz= 1 + S, find the minimum value of the sum x2+ (1 + U)y+ 4z2, where Sisthe third to last digit andUisthe second to last digit of your student number. Determine the corresponding values of x, yandz. (b) Given that x+y+z= 1 + S, find the maximum value of the product xy3z5+W, rounding your result to 3 significant figures, where Sisthe third to last digit andWis the last digit of your student number. Determine the corresponding values of x, yandz.

## Problem 2 20 marks (calculation) + 5 marks (presentation)

A confectionery company produces Kinder Surprise Eggs. Each egg contains a rectangular (“sur- prise”) box that is placed inside a chocolate shell of an ellipsoid shape in such a way that all the box corners touch the shell. Measuring the lengths x, yandzalong the three orthogonal directions in centimeters, the equation of the ellipsoid shell is given by x2+y2 (5 +U)2+z2 (15 + W)2= 1, where Uisthe second to last digit andWisthe last digit of your student number. (a) Assuming that all the box corners touch the shell, use the AM-GM inequality to find the largest volume of the box. Determine further the corresponding box dimensions. You may neglect the thickness of the shell. (b) In addition to the regular boxes, the company also produces a limited exclusive line of luxury boxes which sides are externally covered by a thin layer of gold. The current gold price is 1,608.34 GBP per ounce. Assuming the thickness of the gold layer is 0 .5 +Smicrons, where Sisthe third to last digit of your student number, calculate the cost (in GBP) of such a golden cover that is needed for a box with the maximum volume. 2

## Problem 3 20 marks (calculation) + 5 marks (presentation)

An isosceles triangle ABC with the base BCis inscribed into a downward parabola. The vertex A coincides with the parabola vertex, and the vertices BandCare located at the opposite branches of the parabola. The triangle has the base of length 2(1 + U) and the height of length 2 + W, where Uisthe second to last digit andWisthe last digit of your student number. (a) Make a sketch and find the corresponding equation of the parabola. (b) Suppose that a point Dis chosen on the parabola branch between the points AandB, and the point Eis chosen symmetrically on the other branch. Determine the location of these points which maximizes the area of the pentagon ADBCE and find this area.

## Problem 4 20 marks (calculation) + 5 marks (presentation)

Mr. Smith returned home after some time away and discovered that there was no electricity in his house. To figure out when the electricity went off and whether to complain to the electricity company, he immediately recorded the temperature in his kitchen freezer that was −10◦C. Then two hours later the temperature became ( −9 + 0 .1·W)◦C. The electric power to the house was restored three hours after Mr. Smith’s return. He also knew that normally the temperature in the freezer should be −20◦C. The temperature in the kitchen was (18 + 0 .2·U)◦C all the time. Here Wisthe last digit andUisthe second to last digit of your student number. Estimate for how long the electricity was switched off, assuming two different laws of cooling: (a) the rate of the temperature rise in the freezer is proportional to the difference between the temperature in the kitchen and the temperature in the freezer; (b) the rate of the temperature rise in the freezer is proportional to the power 5/4of the difference between the temperature in the kitchen and the temperature in the freezer. Determine whether Mr. Smith has grounds to complain to the electricity company which has promised to fix any power outages and restore electricity within eight hours maximum. (If the two estimates lead to different conclusions, you may advise him to look for more information, e.g., by speaking to his neighbours.) 3