1. You have been hired as a consultant by the company "Bird Baths R Us” to help determine why one of their products is frequently returned by customers.The CEO expects you to justify your conclusions using both graphical and numerical data. You should aim to be as precise as possible in your analysis.Please upload screenshots of any graphical material as well as any Excel files (not screenshots) you use. The product in question is a hemispherical bird bath known as the "Avosphere."It is ten inches deep and features six perches for visiting birds. The height x, in inches, of the water as the Avosphere is being filled is modeled by the differential equation: \frac{d x}{d t}=\frac{60\left(1-20 k x+k x^{2}\right)}{20 x-x^{2}} where the time t is measured in hours and k is a constant that measures how quickly water evaporates. If there were no evaporation, k would be zero. Your preliminary tests have determined k to be .02. We will assume the bird bath initially has 1 inch of water. • In the "DFIELD Direction Field" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “DFIELD Equation" window, you can change the values in the"Display Window." Use Min t = Min x = 0. You will need to decide what you want the maximum values to be. (a) Using your evaporation constant, what is the height of the water after 2hours? What would the height be after 2 hours if there were no evaporation? (b) Assuming no evaporation, how long until the bird bath is full of water? (d) Based on your answers above, why do you think customers are dissatisfied with this product? (e) Approximately what value would Bird Baths R Us need to reduce theevaporation constant to so that customers can get the Avosphere at least70% full? (c) With your evaporation constant, what is the maximum depth of water acustomer can achieve in their Avosphere?

1. The Simple Linear Regression model can be written as Y₁ = B₂+ B₁2, +4 i=1,2,..., n where 3, and 3, are unknown parameters and e,' s are independent and identically dis- tributed random variables with E(e) = 0 and Var(e) = 0². Also, B, and 3₁ denotes the least-squares estimates for the parameters 3, and 3₁. (a) Derive the formulae for 8, and 3, directly from the sum of squared error (SSE) formula. [3 Marks] (b) Derive the formulae for the variances of the estimators , and 3₁. Express the results in terms of o2. [4 Marks] 2. Data on horse racing were collected. The following table gives the attendance (in thousands of people) at a racetrack and the amount of money (in millions) that was bet on n = 20 randomly selected days. The data is also available in the file Gambling.xlsx. (See Test4_supplementary.txt for instructions to import the Excel data into R.) Attendance Amount Attendance Amount (thousands (millions) (thousands) (millions) 36.5 149.8 50.6 200.2 234.0 174.3 216.1 250.1 201.8 229.4 257.7 311.3 301.7 308.9 339.4 317.4 51.4 50.9 48.8 58.2 61.5 68.5 66.7 71.2 77.0 307.8 349.2 404.5 Page 2 43.9 57.9 54.5 63.2 64.4 67.9 70.9 83.8 79.7 340.4 455.6 410.3

The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?

The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?

1. Consider the model defined by x^{\prime}(t)=x(2-0.4 x-0.3 y) y^{\prime}(t)=y(1-0.1 y-0.3 x) nd and classify the equilibrium points of this system. You can use alculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calcula-tor/computer to help with the computations! (c) Now, use "pplane.jar" and print the phase plane with the nullclines. Inorder to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window”, based on your answers to the previous part. To show the nullclines, go up to "Solution"then "Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0)1.5 and y(0) = 3.5.= (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 1 and y(0) = 1.

Q3: Choose one of the following two problems: 1 (a) Show that the following equality is true: d² dt2 m A¹h(x(t)) = x(t)¹A₁V²h₂(x(t)) [EXX³h (@(0))] # i=1 (t) + A¹Vh(x(t))ä(t) [15 marks] Where A Rmx1, h: Rnx1 → Rmx1, x R → Rx1 and t E R. The bold notation Vh denotes the Jacobian matrix of h. This calculation is used in week 10 lectures notes (along with some similar versions). (b) Independently investigate non-linear least squares. Provide a concise derivation and con- clude by showing that each iteration of the Gauss-Newton algorithm is equivalent to solving a linear system. [20 marks]

1.2 Identify degree of freedom and derive the relationship between structural resisting force and displacement of this structure. ΕΙ m L Figure P1.2 A simple beam for Problems 1.2 to 1.4

12:25 PM Fri Jun 30 SIE340 SUM23 Lecture14 Bieten item 1 2 3 4 5 volume, cubic feet 0.2 0.5 0.2 0.2 0.3 A hiker's knapsack will hold no more than 0.8 cubic feet of canned goods. Some food items that are being considered for inclusion in the knapsack are shown in the table. The hiker wants to maximize the protein content of the items that are selected for inclusion in the knapsack. Quiz grams of protein 12 40 15 20 10 [... O (i) If the hiker choose item 1, he must choose item 2. (ii) If the hiker choose item 3, he cannot choose item 4. (a) Write the decision variables and give the meaning of each decision variable. (b) Write the whole LP model. (objective function, constraints, sign restrictions) (c) Add a constraint to satisfy each of the following cases: 76% Done 25/25

2. Consider the model defined by x^{\prime}(t)=x(1-0.1 x-0.05 y) y^{\prime}(t)=y(1.7-0.1 y-0.15 x) (a) Find and classify the equilibrium points of this system. You can use a calculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calculator/computer to help with the computations! (c) Now, use "pplane.jar” and print the phase plane with the nullclines. In-order to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window", based on your answers to the previous part. To show the nullclines, go up to "Solution"then “Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in-the long run under the initial conditions x(0) = 1 and y(0) = 1. (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 4 and y(0) = 10.