## Questions & Answers

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?

An electronics repair shop has 3 technicians, and can handle 4 types of repairs (jobs) only. Not all the technicians have the skills to perform all repairs, and their wages are different. However, repairs can be split and finished independently by those capable of performing them. Data are given in the tables below. 1. The objective is to minimize the cost of all incoming repairs (60 items). Formulate the problem as an optimization problem, identifying all variables and constraints properly and clearly. Explain as necessary . 2. Write the set of AMPL statements which model this problem (by hand or otherwise)

Let x, y and z be the sides of a rectangular body. 1. Formulate the problem of finding the tetrahedral body of maximum volume with diagonal of unit length. 2. Write the KKT conditions for this problem. 3. Is the cube an optimal solution? Justify your answer.

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. 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.

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.

2. You've been hired by the New York City Office of Public Sanitation to analyze the pests in the sewers. The Chief of Public Sanitation 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 following system of differential equations describes the populations of cock-roaches and rats living in the sewers. x^{\prime}(t)=.05 x-.001 x y y^{\prime}(t)=-.01 y^{2}-.1 y+.02 x y Both species are measured in millions and time is measured in months. • In the "PPLANE Phase Plane" window menu bar, select Options → Delay Time Per Point → 10 Milliseconds Options → Solution Direction → Forward ● In the “PPLANE Equation” window, you can change the values in the"Display Window." Your minimum values should be zero since populations can't be negative. You will need to decide what you want the maximum values to be. • Remember that the command Graph → Both x-t & y-t allows you to seea graph of the solution functions! (a) Which of x or y is the population of cockroaches? Which of x or y is the population of rats? Completely justify your answer. (b) In the absence of one of these two species, describe the growth of the other. (c) Starting with a population of 6 million rats and 4 million cockroaches,when does the number of cockroaches overtake the number of rats? Then,how long does it take for the number of rats to catch back up and overtake the number of cockroaches? What is the minimum rat population during this period of time? (d) Find and classify the equilibrium points of the system. What does this tell you about the long term behavior of the two species? The Chief of Public Sanitation wants to know whether it is more effectiveto poison the cockroaches or rats. What recommendation do you make?Completely justify your answer.