Question:

You are buying a house in Fall Creek. The neighborhood is susceptible to flooding as a result of intense summer thunderstorms and ice jams. During the decade you plan to live in the house, you know there is a chance that a 100-year flood will occur and some chance that a 500-year flood will occur. If a 100-year flood occurs, the damages to the house will be "moderate" ($30,000) with 90% probability and "severe" ($50,000) with 10% probability. If a 500-year flood occurs, the damages to the house will be "moderate" with 70% probability and "severe" with 30% probability. Any other flooding event of smaller magnitude can be assumed to have no damages. The probability of having more than one flood during the period can also be neglected. You can take one of three actions: (1) do not get flood protection (2) install foundation vents, at a cost of $5,000, so that it can protect against moderate impacts, or (3) spend $15,000 and design the house to protect against any severe impacts.

Show that the probability that at least one 100-year flood and at least one 500-yearflood will occur in a decade are approximately 9.6% and 2% respectively. (Hint: a 100-year flood is a flood event that has a 1 in 100 chance (or 1%)of being equaled or exceeded any given year)

Draw a decision tree for this problem. Fully label all branches, nodes, and outcomes with the given information.

Calculate the expected value at each node. If money is the only concern, what should your decision be?

How does the idea of making this decision solely on expected value make you feel? Would you feel comfortable making this real-world decision using a decision tree? Explain why or why not.

E. After discussing with your partner, you decide to install foundation vents and protect your house against moderate impacts. You move in the house and receive a letter from the city proposing a new plan for the residents of Fall Creek to raise the protection in Cascadilla Creek, effectively reducing the impacts of a 500-year flood

6.7 (Reachability matrix for reachable canonical form) Consider a system in reach- able canonical form. Show that the inverse of the reachability matrix is given by al a2 an 0 1 a1 . . an-1 = 0 0 1 . . a1 0 0 0 . . . 6.7 (Reachability matrix for reachable canonical form) Consider a system in reach- able canonical form. Show that the inverse of the reachability matrix is given by a a2 an 0 1 a1 . . an-1 W = 0 0 1 . . a1 0 0 0 . . .A3) Consider the system described by the transfer function. G(S) = 45 + 45+8 252+ 65+ 4 a) (5pts) Write down the controllable canonical form. b) (5pts) Write down the observable canonical form. c) (10pts) Let Xc denotes the state vector for the controllable canonical form and X for the observable canonical form. Find the transformation matrix P such that Xc = PXo. Solution:Problem 1. Solve the differential equation: dx 2 x 3x2 dt with the initial condition x = 1.0 when t = 2.0. Determine the value of x when t = 5.0 and use 100 steps with the following three methods: a) Euler b) Second-order Runge-Kutta c) Fourth-order Runge-Kutta The exact solution is: 25 x(5.0) = 4+ 3In(2) - 3 In (5) Fill in the table below. Give your % Difference values with 3 significant figures. Method x ( 5.0) Difference Euler 2"d-Order Runge-Kutta 4th-Order Runge-Kutta1. compare runge kutta method of order 2 and taylor method of order 2 using following example (runge kutta for w, = 09 19 y'(x) =2ry ,0