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

CE2.2a Use the numerical parameters in Table 2.4 for this and all ensu- ing CE2 assignments (see Figure 1.15). Simulate and plot the open-loop state variable responses for three cases (for this problem use the state-space realizations of CE1.2b); assume zero initial state for all cases [except Case i(b) below]: i. Single-input, single-output: input f () and output O(f). (a) unit impulse input f () and zero initial state. (b) zero input f () and an initial condition of e (0) = 0.1 rad (zero initial conditions on all other state variables). ii. Single-input, multiple-output: impulse input f() and two outputs w(t) and e(t). iii. Multiple-input, multiple-output: two unit step inputs f() and r() and two outputs w() and e(1). Simulate long enough to demonstrate the steady-state behavior. What are the system eigenvalues? Based on these eigenvalues and the physical system, explain the system responses. TABLE 2.4 Numerical Parameters for CE2 System Parameter Value Units Name kg cart mass - kg pendulum mass L 0.75 m pendulum length 9.81 m/s2 gravitational acceleration6.7 (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 . . .Select all strategies that you can use to improve the solution of a first order ordinary differential equation, starting from a base case of Euler's method: Use fourth order Runge Kutta method Use Heun's method Use second order Runge Kutta method O Increase the step size Use first order Runge Kutta method O Decrease the step size