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4 Apr 2018

5. (10 points) Let A= 14 -1 61 2 1 6 . You are given that the eigenvalues of A are 11 = 2, | 2 -1 8 12 = 2, and A3 = 9. (a) Find a basis of the eigenspace corresponding to each eigenvalue. (b) Is A diagonalizable? If so, find an invertible matrix P and a diagonal matrix D such that A = PDP-1; if not, explain why not.

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5 Apr 2018
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3 -97 15) 8(a) Let A Find all the eigenvalues of A and a basis for each of the correspond- 1 -3 ing eigenspaces. (5) 8(b) Suppose B is a 3 x 3 matrix whose eigenvalues are 0, 1, and 2. Is B diagonalizable? Is B invertible? Explain briefly why or why not.

 Mr and Mrs Spin operate a Ferris Wheel at various shows and carnivals. A passenger sits in a capsule and enjoys the ride and views. Below is a diagram showing their magnificent wheel:
Passengers embark and disembark the capsule from the platform (a maximum of 6 passengers are allowed at any given time). The time for a passenger to enter the capsule and return to the platform is 6x minutes. The process repeats until the final passenger(s) embark at no later than 10 pm each evening. The Ferris Wheel runs in an anti-clockwise direction. The height h, in metres, from the ground to a capsule, is measured from the ground to the ball on top of any capsule. It is modelled by 𝒉(𝒕) = 𝒂 − 𝐛cos (𝝅/ 𝟑) A passenger pays 3y dollars for a ride on the wheel. The variables a, b, x and y are as follows:  a is equal to 14 (i.e. a=14)  b is equal to the last digit of your student id plus 2 (e.g. if your id is 4567891 then b=3).  x is equal to the third digit of your id plus 3 (e.g. if your id is 4567891 when x=9).  y is equal to the month that you are born plus 6 (e.g. if you are born in July then y=13). Mr and Mrs Spin would like you to do the following mathematical analysis and present the results in a report for their consumption.
Task 1:
What are the minimum and maximum height of a capsule on the Ferris Wheel?
Task 2:
Determine the heights of each of the capsules on the Ferris Wheel while passengers either embark or disembark a capsule. Present these heights in table form, with appropriate labelling.
Task 3:
Determine the time when the final passenger(s) will disembark the capsule and the Ferris Wheel cease for the evening (remember that the final passengers will embark no later than 10 pm). Note, it is possible that the final passenger(s) may embark the capsule prior to 10 pm, so the answer to this task may not be 10 pm + 6x.

Task 4:
Evaluate the number of capsules that will have a passenger(s) embark for any day, given the Ferris Wheel will commence at 4 pm and the time for passenger(s) to embark and disembark is considered negligible.
Task 5:
On average, 4.8 passengers, on any given day, will embark on a capsule each time. Given the answer to Task 4, find the total number of passengers, on average, that will take a ride on the Ferris Wheel in a day.
Task 6:
Given the answers to Tasks 4 and 5, determine the total ticket sales made on a given day.
Task 7:
Create a set of equations to determine:
1. The number of capsules on a given day could be filled with passengers.
2. The total number of passengers on any given day, for an average.
3. The total ticket sales for any given day, based on the number of capsules per day and an average number of passengers per capsule.
Task 8:
Create a schedule (in table form) for total ticket sales for a day, given average passengers between 4 and 6 inclusive and in increments of 0.2 (i.e. 4, 4.2, 4.4, 4.6, etc.) per capsule.
Task 9:
Report these results in a document for Mr and Mrs Spin. An appendix should show all working out for each of the tasks. This document is electronic in format and no handwriting is acceptable.
 

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